Norm (mathematics)
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In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that satisfies certain properties pertaining to scalability and additivity, and assigns a strictly positive real number to each vector in a vector space over the field of real or complex numbers—except for the zero vector, which is assigned zero.^{[1]} A pseudonorm (seminorm), on the other hand, is allowed to assign zero to some nonzero vectors (in addition to the zero vector).^{[2]}
The term "norm" is commonly used to refer to the vector norm in Euclidean space. It is known as the "Euclidean norm" (see below) which is technically called the L2norm. The Euclidean norm maps a vector to its length in Euclidean space. Because of this, the Euclidean norm is often known as the magnitude.
A vector space on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than one way.
DefinitionEdit
Given a vector space V over a field F of the real numbers or complex numbers , a norm on V is a nonnegativevalued function p: V → with the following properties:^{[3]}
For all a ∈ F and all u, v ∈ V,
 p(u + v) ≤ p(u) + p(v) (being subadditive or satisfying the triangle inequality).
 p(av) = a p(v) (being absolutely homogeneous or absolutely scalable).
 If p(v) = 0 then v = 0 is the zero vector (being positive definite or being pointseparating).
A seminorm on V is a function p : V → with the properties 1 and 2 above. ^{[4]}
Every vector space V with seminorm p induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is defined by:
 p(W + v) = p(v).
Two norms (or seminorms) p and q on a vector space V are equivalent if there exist two real constants c and C, with c > 0, such that
 for every vector v in V, one has that: c q(v) ≤ p(v) ≤ C q(v).
A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm). Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.
NotationEdit
If a norm p : V → is given on a vector space V then the norm of a vector v ∈ V is usually denoted by enclosing it within double vertical lines: ‖v‖ = p(v). 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 v with single vertical lines is also widespread.
In Unicode, the code point of the "double vertical line" character ‖ is U+2016. The double vertical line should not be confused with the "parallel to" symbol, Unicode U+2225 ( ∥ ), which is used in geometry to signify parallel lines and in network theory, various fields of engineering and applied electronics as parallel addition operator. This is usually not a problem because the former is used in parenthesislike fashion, whereas the latter is used as an infix operator. The double vertical line used here should also not be confused with the symbol used to denote lateral clicks in linguistics, Unicode U+01C1 ( ǁ ). The single vertical line  is called "vertical line" in Unicode and its code point is U+007C.
In LaTeX and related markup languages, the macro \
is often used to denote a norm.
ExamplesEdit
 All norms are seminorms.
 The trivial seminorm has p(x) = 0 for all x in V.
 Every linear form f on a vector space defines a seminorm by x → f(x).
Absolutevalue normEdit
The absolute value
is a norm on the onedimensional vector spaces formed by the real or complex numbers.
The absolute value norm is a special case of the L1 norm.
Any norm on a onedimensional vector space is equivalent (up to scaling) to the absolute value norm, meaning that there is a normpreserving isomorphism of vector spaces ; normpreserving means that . This is given by sending to a vector of norm 1.
Proof of uniqueness (up to scaling) in one dimension


In more detail, given a norm on a onedimensional vector space , take a nonzero vector . Then by positivedefiniteness (property 3), so by scaling the vector, is a vector of norm 1, by absolute homogeneity (property 2): . Then the map given by is a normpreserving isomorphism of vector spaces. Note that the map of spaces is not unique; it is only defined up to a scalar of unit norm: could be used instead (over the real numbers, ). 
Euclidean normEdit
On an ndimensional Euclidean space , the intuitive notion of length of the vector x = (x_{1}, x_{2}, ..., x_{n}) 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.^{[5]}
The Euclidean norm is by far the most commonly used norm on , but there are other norms on this vector space as will be shown below. However, all these norms are equivalent in the sense that they all define the same topology.
On an ndimensional complex space the most common norm is
In both cases the norm can be expressed as the square root of the inner product of the vector and itself:
where is represented as a column vector ([x_{1}; x_{2}; ...; x_{n}]), and denotes its conjugate transpose.
This formula is valid for any inner product space, including Euclidean and complex spaces. For Euclidean spaces, the inner product is equivalent to the dot product. Hence, in this specific case the formula can be also written with the following notation:
The Euclidean norm is also called the Euclidean length, L^{2} distance, ℓ^{2} distance, L^{2} norm, or ℓ^{2} norm; see L^{p} space.
The set of vectors in whose Euclidean norm is a given positive constant forms an nsphere.
Euclidean norm of a complex numberEdit
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 x + i y as a vector in the Euclidean plane, makes the quantity (as first suggested by Euler) the Euclidean norm associated with the complex number.
Taxicab norm or Manhattan normEdit
The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x.
The set of vectors whose 1norm 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 _{1} norm. The distance derived from this norm is called the Manhattan distance or _{1} distance.
The 1norm is simply the sum of the absolute values of the columns.
In contrast,
is not a norm because it may yield negative results.
pnormEdit
Let p ≥ 1 be a real number. The norm (also called norm) of vector is
For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches the pnorm approaches the infinity norm or maximum norm:
The pnorm is related to the generalized mean or power mean.
This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm,^{[6]} because it violates the triangle inequality. 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
(without pth root) defines a distance that makes L^{p}(X) into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory, and harmonic analysis. However, outside trivial cases, this topological vector space is not locally convex and has no continuous nonzero linear forms. Thus the topological dual space contains only the zero functional.
The partial derivative of the pnorm is given by
The derivative with respect to x, therefore, is
where denotes Hadamard product and is used for absolute value of each component of the vector.
For the special case of p = 2, this becomes
or
Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)Edit
If is some vector such that , then:
The set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge length 2c.
Zero normEdit
In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the Fspace of sequences with F–norm .^{[7]} Here we mean by Fnorm some realvalued function on an Fspace with distance d, such that . The Fnorm described above is not a norm in the usual sense because it lacks the required homogeneity property.
Hamming distance of a vector from zeroEdit
In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinatewise 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 nonzero point; indeed, the distance from zero remains one as its nonzero 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 componentwise to vectors, the discrete distance from zero behaves like a nonhomogeneous "norm", which counts the number of nonzero components in its vector argument; again, this nonhomogeneous "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 x is simply the number of nonzero 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 pnorms as p approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous. Indeed, it is not even an Fnorm 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^{[who?]} omit Donoho's quotation marks and inappropriately call the numberofnonzeros function the L^{0} norm, echoing the notation for the Lebesgue space of measurable functions.
Other normsEdit
Other norms on can be constructed by combining the above; for example
is a norm on .
For any norm and any injective linear transformation A we can define a new norm of x, equal to
In 2D, with A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. In 2D, each A 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 1norm (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 centrallysymmetric convex body in (centered at zero) defines a norm on .
All the above formulas also yield norms on without modification.
There are also norms on spaces of matrices (with real or complex entries), the socalled matrix norms.
Infinitedimensional caseEdit
The generalization of the above norms to an infinite number of components leads to ℓ^{ p} and L^{ p} spaces, with norms
for complexvalued sequences and functions on respectively, which can be further generalized (see Haar measure).
Any inner product induces in a natural way the norm
Other examples of infinitedimensional normed vector spaces can be found in the Banach space article.
PropertiesEdit
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1norm the unit circle is a square, for the 2norm (Euclidean norm) it is the wellknown unit circle, while for the infinity norm it is a different square. For any pnorm it is a superellipse (with congruent axes). See the accompanying illustration. Due to the definition of the norm, the unit circle must be convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and for a pnorm).
In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors is said to converge in norm to if as . Equivalently, the topology consists of all sets that can be represented as a union of open balls.
Two norms ‖•‖_{α} and ‖•‖_{β} on a vector space V are called equivalent if there exist positive real numbers C and D such that for all x in V
For instance, on , if p > r > 0, then
In particular,
i.e.,
If the vector space is a finitedimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinitedimensional 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.
Every (semi)norm is a sublinear function, which implies that every norm is a convex function. As a result, finding a global optimum of a normbased objective function is often tractable.
Given a finite family of seminorms p_{i} on a vector space the sum
is again a seminorm.
For any norm p on a vector space V, we have that for all u and v ∈ V:
 p(u ± v) ≥ p(u) − p(v).
Proof: Applying the triangular inequality to both and :
Thus, p(u ± v) ≥ p(u) − p(v).
If and are normed spaces and is a continuous linear map, then the norm of and the norm of the transpose of are equal.^{[8]}
For the L^{p} norms, we have Hölder's inequality^{[9]}
A special case of this is the Cauchy–Schwarz inequality:^{[9]}
Classification of seminorms: absolutely convex absorbing setsEdit
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 p_{A} called the gauge of A, defined as
 p_{A}(x) := inf{α : α > 0, x ∈ αA}
with the property that
 {x : p_{A}(x) < 1} ⊆ A ⊆ {x : p_{A}(x) ≤ 1}.
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:
 Suppose now that (p) contains a single p: since (p) is separating, p is a norm, and A = {p < 1} is its open unit ball. Then A is an absolutely convex bounded neighbourhood of 0, and p = p_{A} is continuous.
 The converse is due to Andrey Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely:
 If V is an absolutely convex bounded neighbourhood of 0, the gauge g_{V} (so that V = {g_{V} < 1}) is a norm.
GeneralizationsEdit
There are several generalizations of norms and seminorms. If p is absolute homogeneity but in place of subadditivity we require that
2′.  there is a such that for all 
then p satisfies the triangle inequality but is called a quasiseminorm and the smallest value of b for which this holds is called the multiplier of p; if in addition p separates points then it is called a quasinorm.
On the other hand, if p satisfies the triangle inequality but in place of absolute homogeneity we require that
1′.  there exists a k such that and for all and scalars : 
then p is called a kseminorm.
We have the following relationship between quasiseminorms and kseminorms:
 Suppose that q is a quasiseminorm on a vector space X with multiplier b. If then there exists kseminorm p on X equivalent to q.
The concept of norm in composition algebras does not share the usual properties of a norm. A composition algebra (A, *, N) consists of an algebra over a field A, an involution *, and a quadratic form N, which is called the "norm". In several cases N is an isotropic quadratic form so that A has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.
See alsoEdit
NotesEdit
 ^ "Encyclopedia of Mathematics".
 ^ Knapp, A.W. (2005). Basic Real Analysis. Birkhäuser. p. [1]. ISBN 9780817632502.
 ^ Pugh, C.C. (2015). Real Mathematical Analysis. Springer. p. page 28. ISBN 9783319177700. Prugovečki, E. (1981). Quantum Mechanics in Hilbert Space. p. page 20.
 ^ Rudin, W. (1991). Functional Analysis. p. 25.
 ^ Chopra, Anil (2012). Dynamics of Structures, 4th Ed. PrenticeHall. ISBN 0132858037.
 ^ Except in , where it coincides with the Euclidean norm, and , where it is trivial.
 ^ Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi, 524, doi:10.1007/9789401577588, ISBN 9027721866, MR 0920371, OCLC 13064804
 ^ Treves pp. 242–243
 ^ ^{a} ^{b} Golub, Gene; Van Loan, Charles F. (1996). Matrix Computations (Third ed.). Baltimore: The Johns Hopkins University Press. p. 53. ISBN 080185413X.
ReferencesEdit
 Bourbaki, Nicolas (1987). "Chapters 1–5". Topological vector spaces. Springer. ISBN 3540136274.
 Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN 012566060X.
 Trèves, François (1995). Topological Vector Spaces, Distributions and Kernels. Academic Press, Inc. pp. 136–149, 195–201, 240–252, 335–390, 420–433. ISBN 0486453529.
 Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. SpringerVerlag. pp. 3–5. ISBN 9783540115656. Zbl 0482.46002.