Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.
A pseudonorm or seminorm satisfies the first two properties of a norm, but may be zero for other vectors than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a "seminormed vector space".
Definition.
Given a vector space formula_1 over a subfield of the complex numbers formula_2, a norm on is a real-valued function formula_3 with the following properties, where formula_4 denotes the usual absolute value of a scalar formula_5:
A seminorm on formula_1 is a function formula_3 that has properties (1) and (2) so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1) and (2) imply that if formula_17 is a norm (or more generally, a seminorm) then formula_18 and that formula_17 also has the following property:
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 . Then and are called equivalent, if there exist two real constants and with such that for every vector ,
The norms and are equivalent if and only if they induce the same topology on . Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.
Notation.
If a norm formula_21 is given on a vector space , then the norm of a vector is usually denoted by enclosing it within double vertical lines: Such notation is also sometimes used if is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation with single vertical lines is also widespread.
In LaTeX and related markup languages, the double bar of norm notation is entered with the macro codice_1, which renders as The double vertical line used to denote parallel lines, parallel operator and parallel addition is entered with codice_2 and is rendered as Although looking similar, these two macros must not be confused as codice_1 denotes a bracket and codice_2 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 codice_5 when used as a bracket, and as codice_6 when used as an operator.
In Unicode, the code point of the "double vertical line" character ‖ is U+2016. The "double vertical line" symbol should not be confused with the "parallel to" symbol, Unicode U+2225 ( ∥ ), which is intended to denote parallel lines and parallel operators. The double vertical line should also not be confused with Unicode U+01C1 ( ǁ ), aimed to denote lateral clicks in linguistics.
The single vertical line | is called "vertical line" in Unicode and its code point is U+007C.
Examples.
Every (real or complex) vector space admits a norm: If is a Hamel basis for a vector space then the real-valued map that sends (where all but finitely many of the scalars are 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
is a norm on the one-dimensional vector spaces formed by the real or complex numbers.
Any norm on a one-dimensional vector space is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces where formula_23 is either formula_24 or and norm-preserving means that 
This isomorphism is given by sending formula_25 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 -dimensional Euclidean space the intuitive notion of length of the vector is captured by the formula
This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem. 
This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares.
The Euclidean norm is by far the most commonly used norm on formula_27, but there are other norms on this vector space as will be shown below. 
However, all these norms are equivalent in the sense that they all define the same topology.
The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis. 
Hence, the Euclidean norm can be written in a coordinate-free way as
The Euclidean norm is also called the L"2 norm, ℓ"2 norm, 2-norm, or square norm; see "L""p" space. 
It defines a distance function called the Euclidean length, L"2 distance, or ℓ"2 distance.
The set of vectors in formula_29 whose Euclidean norm is a given positive constant forms an -sphere.
Euclidean norm of complex numbers.
The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane This identification of the complex number as a vector in the Euclidean plane, makes the quantity formula_30 (as first suggested by Euler) the Euclidean norm associated with the complex number.
Quaternions and octonions.
There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers formula_31, the complex numbers formula_32, the quaternions formula_33, and lastly the octonions formula_34, where the dimensions of these spaces over the real numbers are , , , and , respectively. 
The canonical norms on formula_31 and formula_32 are their absolute value functions, as discussed previously.
The canonical norm on formula_33 of quaternions is defined by
for every quaternion formula_39 in formula_33. This is the same as the Euclidean norm on formula_33 considered as the vector space Similarly, the canonical norm on the octonions is the just Euclidean norm on 
On an -dimensional complex space the most common norm is
In this case, the norm can be expressed as the square root of the inner product of the vector and itself:
where formula_44 is represented as a column vector and formula_45 denotes its conjugate transpose.
This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation:
Taxicab norm or Manhattan norm.
The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point .
The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope of dimension equivalent to that of the norm minus 1. 
The Taxicab norm is also called the ℓ"1 norm. The distance derived from this norm is called the Manhattan distance or ℓ"1 distance.
The 1-norm is simply the sum of the absolute values of the columns.
In contrast,
is not a norm because it may yield negative results.
"p"-norm.
Let be a real number. 
The -norm (also called formula_49-norm) of vector formula_50 is
For , we get the taxicab norm, for , we get the Euclidean norm, and as approaches the -norm approaches the infinity norm or :
The -norm is related to the generalized mean or power mean.
This definition is still of some interest for , but the resulting function does not define a norm, because it violates the triangle inequality. 
What is true for this case of , even in the measurable analog, is that the corresponding class is a vector space, and it is also true that the function
(without th root) defines a distance that makes into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic analysis.
However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional.
The partial derivative of the -norm is given by
The derivative with respect to , therefore, is
where denotes Hadamard product and formula_56 is used for absolute value of each component of the vector.
For the special case of , this becomes
or
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm).
If formula_59 is some vector such that then:
The set of vectors whose infinity norm is a given constant, , forms the surface of a hypercube with edge length 2"c".
Zero norm.
In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm formula_61. 
Here we mean by "F-norm" some real-valued function formula_62 on an F-space with distance , such that 
The "F"-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.
Hamming distance of a vector from zero.
In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the "Hamming distance", which is important in coding and information theory. 
In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. 
However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. 
When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.
In signal processing and statistics, David Donoho referred to the "zero" "norm" with quotation marks. 
Following Donoho's notation, the zero "norm" of is simply the number of non-zero coordinates of , or the Hamming distance of the vector from zero. 
When this "norm" is localized to a bounded set, it is the limit of -norms as approaches 0. 
Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous. 
Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument. 
Abusing terminology, some engineers omit Donoho's quotation marks and inappropriately call the number-of-nonzeros function the "L"0 norm, echoing the notation for the Lebesgue space of measurable functions.
Infinite dimensions.
The generalization of the above norms to an infinite number of components leads to and spaces, with norms
for complex-valued sequences and functions on formula_64 respectively, which can be further generalized (see Haar measure).
Any inner product induces in a natural way the norm 
Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article.
Composite norms.
Other norms on formula_27 can be constructed by combining the above; for example
is a norm on 
For any norm and any injective linear transformation we can define a new norm of , equal to
In 2D, with a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram of a particular shape, size, and orientation.
In 3D, this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base).
There are examples of norms that are not defined by "entrywise" formulas. For instance, the Minkowski functional of a centrally-symmetric convex body in formula_27 (centered at zero) defines a norm on formula_27 (see below).
All the above formulas also yield norms on formula_70 without modification.
There are also norms on spaces of matrices (with real or complex entries), the so-called matrix norms.
In abstract algebra.
Let be a finite extension of a field of inseparable degree , and let have algebraic closure . If the distinct embeddings of are , then the Galois-theoretic norm of an element is the value formula_71. 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 formula_72 in composition algebras does "not" share the usual properties of a norm as it may be negative or zero for "z" ≠ 0. A composition algebra consists of an algebra over a field "A", an involution *, and a quadratic form which is called the "norm".
The characteristic feature of composition algebras is the homomorphism property of "N": for the product "wz" of two elements "w" and "z" of the composition algebra, its norm satisfies For and O the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In other composition algebras the norm is an isotropic quadratic form.
Properties.
For any norm "p" on a vector space "V", the reverse triangle inequality holds: for all u and ,
If is a continuous linear map between normed space, then the norm of and the norm of the transpose of are equal.
For the Lp norms, we have Hölder's inequality
A special case of this is the Cauchy–Schwarz inequality:
formula_74
Equivalence.
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square, for the 2-norm (Euclidean norm), it is the well-known unit circle, while for the infinity norm, it is a different square. For any "p"-norm, it is a superellipse with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and formula_75 for a "p"-norm).
In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors formula_76 is said to converge in norm to if formula_77 as Equivalently, the topology consists of all sets that can be represented as a union of open balls. If is a normed space then for all .
Two norms ‖•‖"α" and ‖•‖"β" on a vector space "V" 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 "V"
For instance, if "p" > "r" ≥ 1 on then
In particular,
That is, 
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 "V" can be classified in terms of absolutely convex absorbing subsets "A" of "V". To each such subset corresponds a seminorm "pA" called the gauge of "A", defined as
where 'inf' is the infimum, with the property that
Conversely:
Any locally convex topological vector space has a local basis 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 {"p" < 1/"n"} turns the space into a locally convex topological vector space so that every p is continuous.
Such a method is used to design weak and weak* topologies.
norm case: