TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a norm is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from a
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to the nonnegative real numbers that behaves in certain ways like the distance from the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * , a Wolverine comic book mini-series published by Marvel Comics in 2002 * , a 1999 ''Buffy the Vampire Slayer'' comic book series * , a major ''Judge Dred ...
: it with scaling, obeys a form of the , and is zero only at the origin. In particular, the
Euclidean distance In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of a vector from the origin is a norm, called the
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
, or 2-norm, which may also be defined as the square root of the
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
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 mathematics, a normed vector space or normed space is a vector space over the Real number, real or Complex number, complex numbers, on which a Norm (mathematics), norm is defined. A norm is the formalization and the generalization to real vec ...
. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''.

# Definition

Given a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
$X$ over a subfield of the complex numbers $\Complex,$ a norm on $X$ is a
real-valued function Mass measured in grams is a function from this collection of weight to positive number">positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, i ...
$p : X \to \R$ with the following properties, where $, s,$ denotes the usual
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

of a scalar $s$: #
SubadditivityIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
/: $p\left(x + y\right) \leq p\left(x\right) + p\left(y\right)$ for all $x, y \in X.$ # Absolute homogeneity: $p\left(s x\right) = \left, s\ p\left(x\right)$ for all $x \in X$ and all scalars $s.$ #
Positive definitenessIn mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite bilinear form, positive-definite. See, in particular: * Positive-definite bilin ...
/: for all $x \in X,$ if $p\left(x\right) = 0$ then $x = 0.$ #* Because property (2) implies $p\left(0\right) = 0,$ some authors replace property (3) with the equivalent condition: for every $x \in X,$ $p\left(x\right) = 0$ if and only if $x = 0.$ A
seminorm In mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some A ...
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 In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and th ...
). 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 $p\left(0\right) = 0$ and that $p$ also has the following property: #
• Nonnegativity: $p\left(x\right) \geq 0$ 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: $\, z\, = p\left(z\right).$ 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 Latex is a stable dispersion (emulsion An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class of ...

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 The parallel operator (also known as reduced sum, parallel sum or parallel addition) \, (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in electrical e ...
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 A bracket is either of two tall fore- or back-facing punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...

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 Unicode, formally the Unicode Standard, is an information technology standard Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requireme ...

, 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 $x_ = \left\left(x_i\right\right)_$ is a
Hamel basis In mathematics, a Set (mathematics), set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred ...
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 In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

$\, x\, = , x,$ is a norm on the
one-dimensional In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
vector spaces formed by the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
$\R^n,$ the intuitive notion of length of the vector $\boldsymbol = \left\left(x_1, x_2, \ldots, x_n\right\right)$ 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 In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

. 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 In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
of two vectors of a
Euclidean vector space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called p ...
is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of their
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
s over an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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 In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

is the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...
(also called the modulus) of it, if the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
is identified with the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
$\R^2.$ 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 number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. These are the real numbers $\R,$ the complex numbers $\Complex,$ the
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s $\mathbb,$ and lastly the
octonion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s $\mathbb,$ 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 In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

functions, as discussed previously. The canonical norm on $\mathbb$ of
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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
octonion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
of the vector and itself: $\, \boldsymbol\, := \sqrt,$ where $\boldsymbol$ is represented as a
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and th ...
and $\boldsymbol^H$ denotes its
conjugate transpose In mathematics, the conjugate transpose (or Hermitian transpose) of an ''m''-by-''n'' matrix (mathematics), matrix \boldsymbol with complex number, complex entries is the ''n''-by-''m'' matrix obtained from \boldsymbol by taking the transpose and ...
. This formula is valid for any
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the
complex dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. 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 In urban planning Urban planning, also known as regional planning, town planning, city planning, or rural planning, is a technical and political process that is focused on the development and design of land use and the built environment, i ...
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 A taxicab geometry is a form of geometry in which the usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of thei ...

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 $\mathbf = \left(x_1, \ldots, x_n\right)$ 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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
, and as approaches $\infty$ the -norm approaches the infinity norm or
maximum norm frame, The perimeter of the square is the set of points in R2 where the sup norm equals a fixed positive constant. In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions ''f'' defined on ...
: $\left\, \mathbf\right\, _\infty := \max_i \left, x_i\.$ The -norm is related to the
generalized mean In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
or power mean. This definition is still of some interest for , but the resulting function does not define a norm, because it violates the . 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. These spaces are of great interest in
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
,
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
harmonic analysis Harmonic analysis is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
. 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 $\mathbf = \left(x_1, x_2, \ldots ,x_n\right),$ 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

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 function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s and for the
F-space In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysi ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the
discrete metric Discrete in science is the opposite of continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random vari ...
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 In information theory Information theory is the scientific study of the quantification, storage, and communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to ...
'', which is important in
coding Coding may refer to: Computer science * Computer programming, the process of creating and maintaining the source code of computer programs * Line coding, in data storage * Source coding, compression used in data transmission * Coding theory * Chann ...
and
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
. 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 Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

,
David Donoho David Leigh Donoho (born March 5, 1957) is a professor of statistics at Stanford University , mottoeng = "The wind of freedom blows" , type = Private university, Private research university , academic_affiliations = Association of American Un ...
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 function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s.

## 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 In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfré ...
). Any
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
induces in a natural way the norm Other examples of infinite-dimensional normed vector spaces can be found in the
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

of a particular shape, size, and orientation. In 3D, this is similar but different for the 1-norm (
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral tri ...

s) and the maximum norm (
prism A prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...
s with parallelogram base). There are examples of norms that are not defined by "entrywise" formulas. For instance, the
Minkowski functionalIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 $N\left(z\right)$ in
composition algebra In mathematics, a composition algebra over a field (mathematics), field is a Non-associative algebra, not necessarily associative algebra over a field, algebra over together with a Degenerate form, nondegenerate quadratic form that satisfies ...
s 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 In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
''A'', an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...
*, and a
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$N\left(z\right) = zz^*,$ which is called the "norm". The characteristic feature of composition algebras is the
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
property of ''N'': for the product ''wz'' of two elements ''w'' and ''z'' of the composition algebra, its norm satisfies $N\left(wz\right) = N\left(w\right)N\left(z\right).$ 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 formIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. In other composition algebras the norm is an
isotropic quadratic form In mathematics, a quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
.

# 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 In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

of $u$ are equal. For the ''L''''p'' norms, we have
Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality Inequality may refer to: Economics * Attention inequality Attention inequality is a term used to target the inequality of distribution of atte ...
$, \langle x, y \rangle, \leq \, x\, _p \, y\, _q \qquad \frac + \frac = 1.$ A special case of this is the
Cauchy–Schwarz inequality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
: $\left, \langle x, y \rangle\ \leq \, x\, _2 \, y\, _2.$

## Equivalence

The concept of
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

(the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

, for the 2-norm (Euclidean norm), it is the well-known unit
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

, while for the infinity norm, it is a different square. For any ''p''-norm, it is a
superellipse A superellipse, also known as a Lamé curve after Gabriel Lamé Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equation In mathematics Mathematics (from Ancie ...

with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

on the space, and this is a 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of vectors $\$ is said to
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...
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 A ball A ball is a round object (usually spherical, but can sometimes be ovoid An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( p ...
. If $\left(X, \, \cdot\, \right)$ is a normed space then $\, x - y\, = \, x - z\, + \, z - y\, \text x, y \in X \text z \in$
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
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 convexIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
absorbing subsets ''A'' of $X.$ To each such subset corresponds a seminorm ''pA'' called the
gauge Gauge (US: , UK: or ) may refer to: Measurement * Gauge (instrument) A gauge, in science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), o ...

of ''A'', defined as In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
, with the property that In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
has a
local basisIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point is the collection of all Neighbourhood (mathematics), neighbourhoods of the point . Definit ...
consisting of absolutely convex sets. A common method to construct such a basis is to use a family (''p'') of seminorms ''p'' that separates points: the collection of all finite intersections of sets turns the space into a
locally convex topological vector space In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
so that every p is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. Such a method is used to design weak and weak* topologies. norm case: :Suppose now that (''p'') contains a single ''p'': since (''p'') is separating, ''p'' is a norm, and $A = \$ is its open
unit ball Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) i ...
. Then ''A'' is an absolutely convex bounded neighbourhood of 0, and $p = p_A$ is continuous. :The converse is due to
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...
: any locally convex and locally bounded topological vector space is
normable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Precisely: :If $X$ is an absolutely convex bounded neighbourhood of 0, the gauge $g_X$ (so that $X = \$ is a norm.