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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a norm is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
from a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
to the non-negative 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 * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
: it commutes with scaling, obeys a form of the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
, and is zero only at the origin. In particular, the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
of a vector from the origin is a norm, called the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
, or
2-norm 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 z ...
, which may also be defined as the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of a vector with itself. A
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
satisfies 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 a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
. 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". 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 a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
.


Definition

Given a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
X over a subfield F of the complex numbers \Complex, a norm on X is a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real fun ...
p : X \to \R with the following properties, where , s, denotes the usual
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of a scalar s: #
Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
/
Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
: p(x + y) \leq p(x) + p(y) for all x, y \in X. #
Absolute homogeneity In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
: p(s x) = \left, s\ p(x) for all x \in X and all scalars s. #
Positive definiteness In 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. See, in particular: * Positive-definite bilinear form * Positive-definite fun ...
/: 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 In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
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 function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
al). 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: #
  • 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.


    Equivalent norms

    Suppose that p and q are two norms (or seminorms) on a vector space X. Then p and q are called equivalent, if there exist two positive real constants c and C with c > 0 such that for every vector x \in X, c q(x) \leq p(x) \leq C q(x). The relation "p is equivalent to q" is reflexive,
    symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
    (c q \leq p \leq C q implies \tfrac p \leq q \leq \tfrac p), and transitive and thus defines an
    equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
    on 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 \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, as explained below), the notation , x, with single vertical lines is also widespread.


    Examples

    Every (real or complex) vector space admits a norm: If x_ = \left(x_i\right)_ is a
    Hamel basis In mathematics, a 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 to as component ...
    for a vector space X then the real-valued map that sends x = \sum_ s_i x_i \in X (where all but finitely many of the scalars s_i are 0) to \sum_ \left, s_i\ 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

    The
    absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
    \, x\, = , x, is a norm on the
    one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...
    vector spaces formed by the
    real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
    or
    complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
    s. 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-preserving
    isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
    of vector spaces f : \mathbb \to X, where \mathbb is either \R or \Complex, and norm-preserving means that , x, = p(f(x)). This isomorphism is given by sending 1 \isin \mathbb 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

    On the n-dimensional
    Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
    \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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
    . 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, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
    of two vectors of a
    Euclidean vector space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
    is the
    dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
    of their
    coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
    s over an
    orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
    . 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, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
    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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
    is the
    absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
    (also called the modulus) of it, if the
    complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
    is identified with the
    Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
    \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 algebra In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital algebra, unital real numbers, real non-associative algebras endowed with a posi ...
    s over the
    real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
    s. These are the real numbers \R, the complex numbers \Complex, the
    quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
    s \mathbb, and lastly the
    octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
    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 mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
    functions, as discussed previously. The canonical norm on \mathbb of
    quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
    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, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
    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, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
    of the
    inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
    of the vector and itself: \, \boldsymbol\, := \sqrt, where \boldsymbol is represented as a
    column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
    \begin x_1 \; x_2 \; \dots \; x_n \end^ and \boldsymbol^H denotes its
    conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
    . This formula is valid for any
    inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
    , 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'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
    . 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, the grid plan, grid street plan, or gridiron plan is a type of city plan in which streets run at right angles to each other, forming a grid. Two inherent characteristics of the grid plan, frequent intersections and orthogona ...
    (like that of the
    New York New York most commonly refers to: * New York City, the most populous city in the United States, located in the state of New York * New York (state), a state in the northeastern United States New York may also refer to: Film and television * '' ...
    borough of
    Manhattan Manhattan (), known regionally as the City, is the most densely populated and geographically smallest of the five boroughs of New York City. The borough is also coextensive with New York County, one of the original counties of the U.S. state ...
    ) to get from the origin to the point x. The set of vectors whose 1-norm is a given constant forms the surface of a
    cross polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...
    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 or a Manhattan geometry is a geometry whose 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 ...
    or \ell_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 p \geq 1 be a real number. The p-norm (also called \ell_p-norm) of vector \mathbf = (x_1, \ldots, x_n) is \, \mathbf\, _p := \left(\sum_^n \left, x_i\^p\right)^. For p = 1, we get the
    taxicab norm A taxicab geometry or a Manhattan geometry is a geometry whose 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 ...
    , for p = 2 we get the
    Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
    , and as p approaches \infty the p-norm approaches the
    infinity norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...
    or
    maximum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the ...
    : \, \mathbf\, _\infty := \max_i \left, x_i\. The p-norm is related to the
    generalized mean In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). De ...
    or power mean. For p = 2, the \, \,\cdot\,\, _2-norm is even induced by a canonical
    inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
    \langle \,\cdot,\,\cdot\rangle, meaning that \, \mathbf\, _2 = \sqrt for all vectors \mathbf. This inner product can expressed in terms of the norm by using the
    polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then t ...
    . On \ell^2, this inner product is the ' defined by \langle \left(x_n\right)_, \left(y_n\right)_ \rangle_ ~=~ \sum_n \overline y_n while for the space L^2(X, \mu) associated with a
    measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
    (X, \Sigma, \mu), which consists of all
    square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
    s, this inner product is \langle f, g \rangle_ = \int_X \overline g(x)\, \mathrm dx. This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm, because it violates the
    triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
    . 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 pth root) defines a distance that makes L^p(X) into a complete metric
    topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
    . These spaces are of great interest in
    functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
    ,
    probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
    and
    harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
    . 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 \, \mathbf\, _p = \frac . The derivative with respect to x, therefore, is \frac =\frac . where \circ denotes Hadamard product and , \cdot, is used for absolute value of each component of the vector. For the special case of p = 2, this becomes \frac \, \mathbf\, _2 = \frac, or \frac \, \mathbf\, _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, c, forms the surface of a
    hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
    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 and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
    s and for the
    F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. ...
    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 d, such that \lVert x \rVert = d(x,0). 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, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
    , the
    discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
    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, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to chan ...
    '', which is important in coding and
    information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
    . 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 subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
    and
    statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
    ,
    David Donoho David Leigh Donoho (born March 5, 1957) is an American statistician. He is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences. His work includes the develop ...
    referred to the ''zero'' "''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" is not truly a norm, because it is not
    positive homogeneous In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
    . 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 L^0 norm, echoing the notation for the Lebesgue space of
    measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
    s.


    Infinite dimensions

    The generalization of the above norms to an infinite number of components leads to \ell^p and L^p 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 was introduced by Alfréd Haar in 1933, though ...
    ). Any
    inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
    induces in a natural way the norm \, x\, := \sqrt. Other examples of infinite-dimensional normed vector spaces can be found in the
    Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
    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 \R^4. For any norm and any
    injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
    linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
    A we can define a new norm of x, equal to \, A x\, . 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: a
    parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
    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. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
    s) and the maximum norm (
    prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
    s with parallelogram base). There are examples of norms that are not defined by "entrywise" formulas. For instance, the
    Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then ...
    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 mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows ...
    .


    In abstract algebra

    Let E be a
    finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — ...
    of a field k of
    inseparable degree In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...
    p^, and let k have algebraic closure K. If the distinct embeddings of E are \left\_j, then the Galois-theoretic norm of an element \alpha \in E is the value \left(\prod_j \right)^. As that function is homogeneous of degree : k/math>, the Galois-theoretic norm is not a norm in the sense of this article. However, the : k/math>-th root of the norm (assuming that concept makes sense) is a norm.


    Composition algebras

    The concept of norm N(z) in
    composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution c ...
    s does share the usual properties of a norm as it may be negative or zero for z \neq 0. A composition algebra (A, ^*, N) 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 product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
    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 inpu ...
    ^*, and a
    quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
    N(z) = z z^* called the "norm". The characteristic feature of composition algebras is the
    homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
    property of N: for the product w z 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 linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical d ...
    . In other composition algebras the norm is an
    isotropic quadratic form In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector sp ...
    .


    Properties

    For any norm p : X \to \R on a vector space X, the
    reverse triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
    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, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
    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 between integrals and an indispensable tool for the study of spaces. :Theorem (Hölder's inequality). Let be a measure space and let with . ...
    , \langle x, y \rangle, \leq \, x\, _p \, y\, _q \qquad \frac + \frac = 1. A special case of this is the
    Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
    : \left, \langle x, y \rangle\ \leq \, x\, _2 \, y\, _2. Every norm is a
    seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
    and thus satisfies all properties of the latter. In turn, every seminorm is a
    sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
    and thus satisfies all properties of the latter. In particular, every norm is a
    convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
    .


    Equivalence

    The concept of
    unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
    (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, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
    , for the 2-norm (Euclidean norm), it is the well-known unit
    circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
    , 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, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape. In the ...
    with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be
    convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
    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, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
    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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
    of vectors \ is said to
    converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
    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 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
    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 In the mathematical field of topology a uniform isomorphism or is a special isomorphism between uniform spaces that respects uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a un ...
    .


    Classification of seminorms: absolutely convex absorbing sets

    All seminorms on a vector space X can be classified in terms of
    absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull ...
    absorbing subsets A of X. To each such subset corresponds a seminorm p_A called the
    gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, es ...
    of A, defined as infimum 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 each element of S, if such an element exists. Consequently, the term ''greatest low ...
    , with the property that locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
    has a
    local basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
    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 ''Separates'' is the second album by English punk rock band 999, released in 1978. ''Separates'' was released in the United States under the title ''High Energy Plan'', with a different cover and slightly altered track listing; on ''High Energ ...
    : the collection of all finite intersections of sets \ turns the space into a
    locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
    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 ...
    . 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, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
    . Then A is an absolutely convex
    bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
    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 Sovi ...
    : any locally convex and locally bounded topological vector space is
    normable 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 ze ...
    . Precisely: :If X is an absolutely convex bounded neighbourhood of 0, the gauge g_X (so that X = \ is a norm.


    See also

    * * * * * * * * * * * * * * *


    References


    Bibliography

    * * * * * * {{DEFAULTSORT:Norm (Mathematics) Functional analysis Linear algebra