Absolute valueFrom Wikipedia, the free encyclopedia

Chapter 1

Absolute value

For other uses, see Absolute value (disambiguation).In mathematics, the absolute value ormodulus |x| of a real number x is the non-negative value of x without regard

The absolute value of a number may be thought of as its distance from zero.

to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. Forexample, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may bethought of as its distance from zero.Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example,an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical andphysical contexts.

1.1 Terminology and notation

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for thecomplex absolute value,[1][2] and it was borrowed into English in 1866 as the Latin equivalent modulus.[1] The termabsolute value has been used in this sense from at least 1806 in French[3] and 1857 in English.[4] The notation |x|, witha vertical bar on each side, was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value includenumerical value[1] and magnitude.[1]

The same notation is used with sets to denote cardinality; the meaning depends on context.

1.2 Definition and properties

2

1.2. DEFINITION AND PROPERTIES 3

1.2.1 Real numbers

For any real number x the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of thequantity) and is defined as[6]

|x| =

{x, if x ≥ 0

−x, if x < 0

As can be seen from the above definition, the absolute value of x is always either positive or zero, but never negative.From an analytic geometry point of view, the absolute value of a real number is that number’s distance from zeroalong the real number line, and more generally the absolute value of the difference of two real numbers is the distancebetween them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisationof the absolute value of the difference (see “Distance” below).Since the square root notation without sign represents the positive square root, it follows that

which is sometimes used as a definition of absolute value of real numbers.[7]

The absolute value has the following four fundamental properties:

Other important properties of the absolute value include:

Two other useful properties concerning inequalities are:

|a| ≤ b ⇐⇒ −b ≤ a ≤ b

|a| ≥ b ⇐⇒ a ≤ −b or b ≤ a

These relations may be used to solve inequalities involving absolute values. For example:

Absolute value is used to define the absolute difference, the standard metric on the real numbers.

1.2.2 Complex numbers

Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directlygeneralised for a complex number. However the geometric interpretation of the absolute value of a real number as itsdistance from 0 can be generalised. The absolute value of a complex number is defined as its distance in the complexplane from the origin using the Pythagorean theorem. More generally the absolute value of the difference of twocomplex numbers is equal to the distance between those two complex numbers.For any complex number

z = x+ iy,

where x and y are real numbers, the absolute value ormodulus of z is denoted |z| and is given by[8]

|z| =√x2 + y2.

4 CHAPTER 1. ABSOLUTE VALUE

Im

Re

y

−y

0 x

r

r

φ

φ

z=x+iy

z=x−iy

The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complexconjugate z have the same absolute value.

When the imaginary part y is zero this is the same as the absolute value of the real number x.When a complex number z is expressed in polar form as

1.3. ABSOLUTE VALUE FUNCTION 5

z = reiθ

with r ≥ 0 and θ real, its absolute value is

|z| = r

The absolute value of a complex number can be written in the complex analogue of equation (1) above as:

|z| =√z · z

where z is the complex conjugate of z. Notice that, contrary to equation (1):

|z| ̸=√z2

The complex absolute value shares all the properties of the real absolute value given in equations (2)–(11) above.Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolutevalue as an endomorphism of the multiplicative group of the complex numbers.[9]

1.3 Absolute value function

1

2

3

4

−3 −2 −1 1 2 30

y = |x|

The graph of the absolute value function for real numbers

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It ismonotonically decreasing on the interval (−∞,0] and monotonically increasing on the interval [0,+∞). Since a realnumber and its opposite have the same absolute value, it is an even function, and is hence not invertible.Both the real and complex functions are idempotent.It is a piecewise linear, convex function.

6 CHAPTER 1. ABSOLUTE VALUE

x

y

f (| x |)

| f (x ) |

f (x )

Composition of absolute value with a cubic function in different orders

1.3.1 Relationship to the sign function

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum)function returns a number’s sign irrespective of its value. The following equations show the relationship between thesetwo functions:

|x| = x sgn(x),

1.4. DISTANCE 7

or

|x| sgn(x) = x,

and for x ≠ 0,

sgn(x) = |x|x.

1.3.2 Derivative

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0. Its derivative for x≠ 0 is given by the step function[10][11]

d|x|dx

=x

|x|=

{−1 x < 0

1 x > 0.

The subdifferential of |x| at x = 0 is the interval [−1,1].[12]

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violatesthe Cauchy–Riemann equations.[10]

The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalisedfunction, the second derivative may be taken as two times the Dirac delta function.

1.3.3 Antiderivative

The antiderivative (indefinite integral) of the absolute value function is

∫|x|dx =

x|x|2

+ C,

where C is an arbitrary constant of integration.

1.4 Distance

See also: Metric space

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complexnumber is the distance from that number to the origin, along the real number line, for real numbers, or in the complexplane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbersis the distance between them.The standard Euclidean distance between two points

a = (a1, a2, . . . , an)

and

b = (b1, b2, . . . , bn)

8 CHAPTER 1. ABSOLUTE VALUE

in Euclidean n-space is defined as:

√√√√ n∑i=1

(ai − bi)2.

This can be seen to be a generalisation of |a − b|, since if a and b are real, then by equation (1),

|a− b| =√(a− b)2.

While if

a = a1 + ia2

and

b = b1 + ib2

are complex numbers, then

The above shows that the “absolute value” distance for the real numbers or the complex numbers, agrees with thestandard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclideanspaces respectively.The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity ofindiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion ofa distance function as follows:A real valued function d on a set X × X is called a metric (or a distance function) on X, if it satisfies the followingfour axioms:[13]

1.5 Generalizations

1.5.1 Ordered rings

The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is anelement of an ordered ring R, then the absolute value of a, denoted by |a|, is defined to be:[14]

|a| =

{a, if a ≥ 0

−a, if a ≤ 0

where −a is the additive inverse of a, and 0 is the additive identity element.

1.5.2 Fields

Main article: Absolute value (algebra)

1.5. GENERALIZATIONS 9

The fundamental properties of the absolute value for real numbers given in (2)–(5) above, can be used to generalisethe notion of absolute value to an arbitrary field, as follows.A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation)[15] ifit satisfies the following four axioms:

Where 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativeness thatv(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values defined aboveare examples of absolute values for an arbitrary field.If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(a − b), is a metric and the followingare equivalent:

• d satisfies the ultrametric inequality d(x, y) ≤ max(d(x, z), d(y, z)) for all x, y, z in F.

•{v(∑n

k=11): n ∈ N

}is bounded in R.

• v(∑n

k=11)≤ 1 for every n ∈ N.

• v(a) ≤ 1 ⇒ v(1 + a) ≤ 1 for all a ∈ F.

• v(a+ b) ≤ max{v(a), v(b)} for all a, b ∈ F.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwiseit is said to be Archimedean.[16]

1.5.3 Vector spaces

Main article: Norm (mathematics)

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, togeneralise the notion to an arbitrary vector space.A real-valued function on a vector space V over a field F, represented as ‖·‖, is called an absolute value, but moreusually a norm, if it satisfies the following axioms:For all a in F, and v, u in V,

The norm of a vector is also called its length or magnitude.In the case of Euclidean space Rn, the function defined by

∥(x1, x2, . . . , xn)∥ =

√√√√ n∑i=1

x2i

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector spaceR1, the absolute value is a norm, and is the p-norm (see Lp space) for any p. In fact the absolute value is the “only”norm on R1, in the sense that, for every norm ‖·‖ on R1, ‖x‖ = ‖1‖ ⋅ |x|. The complex absolute value is a special caseof the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified withthe Euclidean plane R2.

10 CHAPTER 1. ABSOLUTE VALUE

1.6 Notes[1] Oxford English Dictionary, Draft Revision, June 2008

[2] Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877

[3] Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques prisdans l'espace, p. 105 at Google Books

[4] James Mill Peirce, A Text-book of Analytic Geometry at Google Books. The oldest citation in the 2nd edition of the OxfordEnglish Dictionary is from 1907. The term absolute value is also used in contrast to relative value.

[5] Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25

[6] Mendelson, p. 2.

[7] Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1., p. A5

[8] González, Mario O. (1992). Classical Complex Analysis. CRC Press. p. 19. ISBN 9780824784157.

[9] Lorenz, Falko (2008), Algebra. Vol. II. Fields with structure, algebras and advanced topics, Universitext, New York:Springer, p. 39, doi:10.1007/978-0-387-72488-1, ISBN 978-0-387-72487-4, MR 2371763.

[10] Weisstein, Eric W. Absolute Value. From MathWorld – A Wolfram Web Resource.

[11] Bartel and Sherbert, p. 163

[12] Peter Wriggers, Panagiotis Panatiotopoulos, eds., New Developments in Contact Problems, 1999, ISBN 3-211-83154-1, p.31–32

[13] These axioms are not minimal; for instance, non-negativity can be derived from the other three: 0 = d(a, a) ≤ d(a, b) +d(b, a) = 2d(a, b).

[14] Mac Lane, p. 264.

[15] Shechter, p. 260. This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolutevalue

[16] Shechter, pp. 260–261.

1.7 References• Bartle; Sherbert; Introduction to real analysis (4th ed.), John Wiley & Sons, 2011 ISBN 978-0-471-43331-6.

• Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-1.

• Mac Lane, Saunders, Garrett Birkhoff, Algebra, AmericanMathematical Soc., 1999. ISBN 978-0-8218-1646-2.

• Mendelson, Elliott, Schaum’s Outline of Beginning Calculus, McGraw-Hill Professional, 2008. ISBN 978-0-07-148754-2.

• O'Connor, J.J. and Robertson, E.F.; “Jean Robert Argand”.

• Schechter, Eric; Handbook of Analysis and Its Foundations, pp. 259–263, “Absolute Values”, Academic Press(1997) ISBN 0-12-622760-8.

1.8 External links• Hazewinkel, Michiel, ed. (2001), “Absolute value”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• absolute value at PlanetMath.org.

• Weisstein, Eric W., “Absolute Value”, MathWorld.

Chapter 2

Norm (mathematics)

This article is about linear algebra and analysis. For field theory, see Field norm. For ideals, see Ideal norm. Forgroup theory, see Norm (group). For norms in descriptive set theory, see prewellordering.

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictlypositive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zerovector).A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formaldefinition below.A simple example is the 2-dimensional Euclidean space R2 equipped with the Euclidean norm. Elements in thisvector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at theorigin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean normis 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 seminormis called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than oneway.

2.1 Definition

Given a vector space V over a subfield F of the complex numbers, a norm on V is a function p: V → R with thefollowing properties:[1]

For all a ∈ F and all u, v ∈ V,

1. p(av) = |a| p(v), (absolute homogeneity or absolute scalability).

2. p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).

3. If p(v) = 0 then v is the zero vector (separates points).

By the first axiom, absolute homogeneity, we have p(0) = 0 and p(−v) = p(v), so that by the triangle inequality

p(v) ≥ 0 (non-negativity).

A seminorm on V is a function p : V → R with the properties 1. and 2. above.Every vector space V with seminorm p induces a normed space V/W, called the quotient space, where W is thesubspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is clearly well-defined andis given by:

p(W + v) = p(v).

11

12 CHAPTER 2. NORM (MATHEMATICS)

Two norms (or seminorms) p and q on a vector space V are equivalent if there exist two real constants c and C, withc > 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 anorm (seminorm).

2.2 Notation

If a norm p : V → R is given on a vector space V then the norm of a vector v ∈ V is usually denoted by enclosing itwithin 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 codepoint of the “double vertical line” character ‖ is U+2016. The double vertical line should notbe confused with the “parallel to” symbol, Unicode U+2225 ( ∥ ). This is usually not a problem because the formeris used in parenthesis-like fashion, whereas the latter is used as an infix operator. The double vertical line used hereshould also not be confused with the symbol used to denote lateral clicks, Unicode U+01C1 ( ǁ ). The single verticalline | is called “vertical line” in Unicode and its codepoint is U+007C.

2.3 Examples

• 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)|.

2.3.1 Absolute-value norm

The absolute value

∥x∥ = |x|

is a norm on the one-dimensional vector spaces formed by the real or complex numbers.

2.3.2 Euclidean norm

Main article: Euclidean distance

On an n-dimensional Euclidean space Rn, the intuitive notion of length of the vector x = (x1, x2, ..., xn) is capturedby the formula

∥x∥ :=√x21 + · · ·+ x2

n.

This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. TheEuclidean norm is by far the most commonly used norm on Rn, but there are other norms on this vector space as willbe shown below. However all these norms are equivalent in the sense that they all define the same topology.On an n-dimensional complex space Cn the most common norm is

2.3. EXAMPLES 13

∥z∥ :=

√|z1|2 + · · ·+ |zn|2 =

√z1z̄1 + · · ·+ znz̄n.

In both cases we can also express the norm as the square root of the inner product of the vector and itself:

∥x∥ :=√x∗ x,

where x is represented as a column vector ([x1; x2; ...; xn]), and x∗ 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 withthe following notation:

∥x∥ :=√x · x.

The Euclidean norm is also called the Euclidean length, L2 distance, ℓ2 distance, L2 norm, or ℓ2 norm; see Lpspace.The set of vectors in Rn+1 whose Euclidean norm is a given positive constant forms an n-sphere.

Euclidean norm of a complex number

The Euclidean norm of a complex number is the absolute value (also called themodulus) of it, if the complex plane isidentified with the Euclidean plane R2. This identification of the complex number x + iy as a vector in the Euclideanplane, makes the quantity

√x2 + y2 (as first suggested by Euler) the Euclidean norm associated with the complex

number.

2.3.3 Taxicab norm or Manhattan norm

Main article: Taxicab geometry

∥x∥1 :=

n∑i=1

|xi| .

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 1-norm is a given constant forms the surface of a cross polytope of dimension equivalentto that of the norm minus 1. The Taxicab norm is also called the ℓ 1 norm. The distance derived from this norm iscalled the Manhattan distance or ℓ 1 distance.The 1-norm is simply the sum of the absolute values of the columns.In contrast,

n∑i=1

xi

is not a norm because it may yield negative results.

2.3.4 p-norm

Main article: Lp space

Let p ≥ 1 be a real number.

14 CHAPTER 2. NORM (MATHEMATICS)

∥x∥p :=

( n∑i=1

|xi|p)1/p

.

For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches∞ the p-norm approachesthe infinity norm or maximum norm. The p-norm 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,[2] becauseit violates the triangle inequality. What is true for this case of 0 < p < 1, even in the measurable analog, is that thecorresponding Lp class is a vector space, and it is also true that the function

∫X

|f(x)− g(x)|p dµ

(without pth root) defines a distance that makes Lp(X) into a complete metric topological vector space. These spacesare 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 topologicaldual space contains only the zero functional.The derivative of the p-norm is given by

∂

∂xk∥x∥p =

xk |xk|p−2

∥x∥p−1p

.

The derivative with respect to x, therefore, is

∂∥x∥p∂x =

x ◦ |x|p−2

∥x∥p−1p

.

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

∂

∂xk∥x∥2 =

xk

∥x∥2,

or

∂

∂x ∥x∥2 =x

∥x∥2.

2.3.5 Maximumnorm (special case of: infinity norm, uniformnorm, or supremumnorm)

Main article: Maximum norm

∥x∥∞ := max (|x1| , . . . , |xn|) .

The set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge length 2c.

2.3.6 Zero norm

In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurablefunctions and for the F-space of sequences with F–norm (xn) 7→

∑n 2

−nxn/(1 + xn) , which is discussed byStefan Rolewicz in Metric Linear Spaces.[3] Here we mean by F-norm some real-valued function ∥ · ∥ on an F-spacewith distance d, such that ∥x∥ = d(x, 0) .

2.3. EXAMPLES 15

x ∞

∥x∥∞ = 1

Hamming distance of a vector from zero

See also: Hamming distance and discrete metric

In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When appliedcoordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which isimportant in coding and information theory. In the field of real or complex numbers, the distance of the discretemetric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zeroargument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of anorm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discretedistance from zero behaves like a non-homogeneous “norm”, which counts the number of non-zero components inits vector argument; again, this non-homogeneous “norm” is discontinuous.In signal processing and statistics, David Donoho referred to the zero "norm" with quotation marks. FollowingDonoho’s notation, the zero “norm” of x is simply the number of non-zero coordinates of x, or the Hamming distanceof the vector from zero. When this “norm” is localized to a bounded set, it is the limit of p-norms as p approaches 0. Ofcourse, the zero “norm” is not truly a norm, because it is not positive homogeneous. Indeed, it is not even an F-norm inthe sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–

16 CHAPTER 2. NORM (MATHEMATICS)

vector multiplication and with respect to its vector argument. Abusing terminology, some engineers omit Donoho’squotation marks and inappropriately call the number-of-nonzeros function the L0 norm, echoing the notation for theLebesgue space of measurable functions.

2.3.7 Other norms

Other norms on Rn can be constructed by combining the above; for example

∥x∥ := 2 |x1|+√

3 |x2|2 +max(|x3| , 2 |x4|)2

is a norm on R4.For any norm and any injective linear transformation A we can define a new norm of x, equal to

∥Ax∥ .

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 1-norm(octahedrons) and the maximum norm (prisms with parallelogram base).All the above formulas also yield norms on Cn without modification.

2.3.8 Infinite-dimensional case

The generalization of the above norms to an infinite number of components leads to the Lp spaces, with norms

∥x∥p =

(∑i∈N

|xi|p)1/p

resp. ∥f∥p,X =

(∫X

|f(x)|p dx)1/p

(for complex-valued sequences x resp. functions f defined on X ⊂ R ), which can be further generalized (see Haarmeasure).Any inner product induces in a natural way the norm ∥x∥ :=

√⟨x, x⟩.

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

2.4 Properties

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm the unitcircle in R2 is a square, for the 2-norm (Euclidean norm) it is the well-known unit circle, while for the infinity normit 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 is always convex and centrally symmetric (therefore, for example,the unit ball may be a rectangle but cannot be a triangle).In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology preciselywhen 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 opensets. A sequence of vectors {vn} is said to converge in norm to v if ∥vn − v∥ → 0 as n → ∞ . Equivalently, thetopology 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 suchthat for all x in V

C ∥x∥α ≤ ∥x∥β ≤ D ∥x∥α .

2.4. PROPERTIES 17

For instance, on Cn , if p > r > 0, then

∥x∥p ≤ ∥x∥r ≤ n(1/r−1/p) ∥x∥p .

In particular,

∥x∥2 ≤ ∥x∥1 ≤√n ∥x∥2

∥x∥∞ ≤ ∥x∥2 ≤√n ∥x∥∞

∥x∥∞ ≤ ∥x∥1 ≤ n ∥x∥∞ .

If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in thecase 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 bedistinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformlyisomorphic.Every (semi)-norm is a sublinear function, which implies that every norm is a convex function. As a result, finding aglobal optimum of a norm-based objective function is often tractable.Given a finite family of seminorms pi on a vector space the sum

p(x) :=

n∑i=0

pi(x)

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 p(u− 0) and p(v − 0) :

p(u− 0) ≤ p(u− v) + p(v − 0) ⇒ p(u− v) ≥ p(u)− p(v)

p(u− 0) ≤ p(u+ v) + p(0− v) ⇒ p(u+ v) ≥ p(u)− p(v)

p(v − 0) ≤ p(u− v) + p(u− 0) ⇒ p(u− v) ≥ p(v)− p(u)

p(v − 0) ≤ p(u+ v) + p(0− u) ⇒ p(u+ v) ≥ p(v)− p(u)

Thus, p(u ± v) ≥ |p(u) − p(v)|.

If X and Y are normed spaces and u : X → Y is a continuous linear map, then the norm of u and the norm of thetranspose of u are equal.[4]

For the lp norms, we have Hölder’s inequality[5]

∣∣xTy∣∣ ≤ ∥x∥p ∥y∥q

1

p+

1

q= 1.

A special case of this is the Cauchy–Schwarz inequality:[5]

∣∣xTy∣∣ ≤ ∥x∥2 ∥y∥2 .

18 CHAPTER 2. NORM (MATHEMATICS)

2.5 Classification of seminorms: absolutely convex absorbing sets

All seminorms on a vector space V can be classified in terms of absolutely convex absorbing sets in V. To each suchset, A, corresponds a seminorm pA called the gauge of A, defined as

pA(x) := inf{α : α > 0, x ∈ αA}

with the property that

{x : pA(x) < 1} ⊆ A ⊆ {x : pA(x) ≤ 1}.

Conversely:Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common methodto construct such a basis is to use a family (p) of seminorms p that separates points: the collection of all finite inter-sections 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 openunit ball. Then A is an absolutely convex bounded neighbourhood of 0, and p = pA is continuous.

The converse is due to Kolmogorov: any locally convex and locally bounded topological vector space isnormable. Precisely:If V is an absolutely convex bounded neighbourhood of 0, the gauge gV (so that V = {gV < 1}) is anorm.

2.6 Generalizations

There are several generalizations of norms and semi-norms. If p is absolute homogeneity but in place of subadditivitywe require thatthen p satisfies the triangle inequality but is called a quasi-seminorm and the smallest value of b for which this holdsis called themultiplier of p; if in addition p separates points then it is called a quasi-norm.On the other hand, if p satisfies the triangle inequality but in place of absolute homogeneity we require thatthen p is called a k-seminorm.We have the following relationship between quasi-seminorms and k-seminorms:

Suppose that q is a quasi-seminorm on a vector space X with multiplier b. If 0 < k < log22 b then thereexists k-seminorm p on X equivalent to q.

2.7 See also• Normed vector space

• Asymmetric norm

• Matrix norm

• Gowers norm

• Mahalanobis distance

• Manhattan distance

• Relation of norms and metrics

2.8. NOTES 19

2.8 Notes[1] Prugovečki 1981, page 20

[2] Except in R1, where it coincides with the Euclidean norm, and R0, where it is trivial.

[3] Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (EastEuropean Series) 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.;PWN—Polish Scientific Publishers, pp. xvi,524, ISBN 90-277-2186-6, MR 920371, OCLC 13064804

[4] Treves pp. 242–243

[5] Golub, Gene; Van Loan, Charles F. (1996). Matrix Computations (Third ed.). Baltimore: The Johns Hopkins UniversityPress. p. 53. ISBN 0-8018-5413-X.

2.9 References• Bourbaki, Nicolas (1987). “Chapters 1–5”. Topological vector spaces. Springer. ISBN 3-540-13627-4.

• Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN0-12-566060-X.

• Trèves, François (1995). Topological Vector Spaces, Distributions and Kernels. Academic Press, Inc. pp.136–149, 195–201, 240–252, 335–390, 420–433. ISBN 0-486-45352-9.

• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics 936.Springer-Verlag. pp. 3–5. ISBN 978-3-540-11565-6. Zbl 0482.46002.

20 CHAPTER 2. NORM (MATHEMATICS)

x 1

x 2

x ∞

Illustrations of unit circles in different norms.

Chapter 3

Sign (mathematics)

Not to be confused with the sine function in trigonometry.For symbols named "… sign”, see List of mathematical symbols.In mathematics, the concept of sign originates from the property of every non-zero real number to be positive

The plus and minus symbols are used to show the sign of a number.

or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero. Along itsapplication to real numbers, “change of sign” is used throughout mathematics and physics to denote the additiveinverse (multiplication to −1), even for quantities which are not real numbers (so, which are not prescribed to beeither positive, negative, or zero). Also, the word “sign” can indicate aspects of mathematical objects that resemblepositivity and negativity, such as the sign of a permutation (see below).

21

22 CHAPTER 3. SIGN (MATHEMATICS)

3.1 Sign of a number

Every number has multiple attributes (such as value, sign and magnitude). A real number is said to be positive if itsvalue (not its magnitude) is greater than zero, and negative if it is less than zero. The attribute of being positive ornegative is called the sign of the number. Zero itself is not considered to have a sign (though this is context dependent,see below). Also, signs are not defined for complex numbers, although the argument generalizes it in some sense.In common numeral notation (which is used in arithmetic and elsewhere), the sign of a number is often denoted byplacing a plus sign or a minus sign before the number. For example, +3 denotes “positive three”, and −3 denotes“negative three”. When no plus or minus sign is given, the default interpretation is that a number is positive. Becauseof this notation, as well as the definition of negative numbers through subtraction, the minus sign is perceived to havea strong association with negative numbers (of the negative sign). Likewise, "+" associates with positivity.In algebra, a minus sign is usually thought of as representing the operation of additive inverse (sometimes callednegation), with the additive inverse of a positive number being negative and the additive inverse of a negative numberbeing positive. In this context, it makes sense to write −(−3) = +3.Any non-zero number can be changed to a positive one using the absolute value function. For example, the absolutevalue of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| = 3 and |3| = 3.

3.1.1 Sign of zero

The number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote thesame number 0, which is the additive inverse of itself.Note that this definition is culturally determined. In France and Belgium, 0 is said to be both positive and negative.The positive resp. negative numbers without zero are said to be “strictly positive” resp. “strictly negative”.In some contexts, such as signed number representations in computing, it makes sense to consider signed versions ofzero, with positive zero and negative zero being different numbers (see signed zero).One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits. This notation refersto the behaviour of a function as the input variable approaches 0 from positive or negative values respectively; thesebehaviours are not necessarily the same.

3.1.2 Terminology for signs

Because zero is neither positive nor negative (in most countries), the following phrases are sometimes used to referto the sign of an unknown number:

• A number is positive if it is greater than zero.

• A number is negative if it is less than zero.

• A number is non-negative if it is greater than or equal to zero.

• A number is non-positive if it is less than or equal to zero.

Thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero. Forexample, the absolute value of a real number is always non-negative, but is not necessarily positive.The same terminology is sometimes used for functions that take real or integer values. For example, a function wouldbe called positive if all of its values are positive, or non-negative if all of its values are non-negative.

3.1.3 Sign convention

Main article: Sign convention

In many contexts the choice of sign convention (which range of values is considered positive and which negative) isnatural, whereas in others the choice is arbitrary subject only to consistency, the latter necessitating an explicit signconvention.

3.2. SIGN FUNCTION 23

3.2 Sign function

1

−1

y

x

Signum function y = sgn(x)

Main article: Sign function

The sign function or signum function is sometimes used to extract the sign of a number. This function is usuallydefined as follows:

sgn(x) =

−1 ifx < 0,

0 ifx = 0,

1 ifx > 0.

Thus sgn(x) is 1 when x is positive, and sgn(x) is −1 when x is negative. For nonzero values of x, this function canalso be defined by the formula

sgn(x) = x

|x|=

|x|x

where |x| is the absolute value of x.

3.3 Meanings of sign

24 CHAPTER 3. SIGN (MATHEMATICS)

Measuring from the x-axis, angles on the unit circle count as positive in the counterclockwise direction, and negative in the clockwisedirection.

3.3.1 Sign of an angle

In many contexts, it is common to associate a sign with the measure of an angle, particularly an oriented angle or anangle of rotation. In such a situation, the sign indicates whether the angle is in the clockwise or counterclockwisedirection. Though different conventions can be used, it is common in mathematics to have counterclockwise anglescount as positive, and clockwise angles count as negative.It is also possible to associate a sign to an angle of rotation in three dimensions, assuming the axis of rotation has beenoriented. Specifically, a right-handed rotation around an oriented axis typically counts as positive, while a left-handedrotation counts as negative.

3.3.2 Sign of a change

When a quantity x changes over time, the change in the value of x is typically defined by the equation

∆x = xfinal − xinitial.

Using this convention, an increase in x counts as positive change, while a decrease of x counts as negative change.In calculus, this same convention is used in the definition of the derivative. As a result, any increasing function haspositive derivative, while a decreasing function has negative derivative.

3.3. MEANINGS OF SIGN 25

3.3.3 Sign of a direction

In analytic geometry and physics, it is common to label certain directions as positive or negative. For a basic example,the number line is usually drawn with positive numbers to the right, and negative numbers to the left:

As a result, when discussing linear motion, displacement or velocity to the right is usually thought of as being positive,while similar motion to the left is thought of as being negative.On the Cartesian plane, the rightward and upward directions are usually thought of as positive, with rightward beingthe positive x-direction, and upward being the positive y-direction. If a displacement or velocity vector is separatedinto its vector components, then the horizontal part will be positive for motion to the right and negative for motion tothe left, while the vertical part will be positive for motion upward and negative for motion downward.

3.3.4 Signedness in computing

Main article: Signedness

In computing, an integer value may be either signed or unsigned, depending on whether the computer is keeping trackof a sign for the number. By restricting an integer variable to non-negative values only, one more bit can be used forstoring the value of a number. Because of the way integer arithmetic is done within computers, the sign of a signedinteger variable is usually not stored as a single independent bit, but is instead stored using two’s complement or someother signed number representation.In contrast, real numbers are stored andmanipulated as Floating point values. The floating point values are representedusing three separate values, mantissa, exponent, and, sign. Given this separate sign bit, it is possible to represent bothpositive and negative zero. Most programming languages normally treat positive zero and negative zero as equivalentvalues, albeit, they provide means by which the distinction can be detected.

3.3.5 Other meanings

In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematicsand the sciences:

• Words up to sign mean that for a quantity q is known that either q = Q or q = −Q for certain Q. It is oftenexpressed as q = ±Q. For real numbers, it means that only the absolute value |q| of the quantity is known. Forcomplex numbers and vectors, a quantity known up to sign is a stronger condition than a quantity with knownmagnitude: aside Q and −Q, there are many other possible values of q such that |q| = |Q|.

• The sign of a permutation is defined to be positive if the permutation is even, and negative if the permutationis odd.

• In graph theory, a signed graph is a graph in which each edge has been marked with a positive or negative sign.

• In mathematical analysis, a signed measure is a generalization of the concept of measure in which the measureof a set may have positive or negative values.

• In a signed-digit representation, each digit of a number may have a positive or negative sign.

• The ideas of signed area and signed volume are sometimes used when it is convenient for certain areas orvolumes to count as negative. This is particularly true in the theory of determinants.

• In physics, any electric charge comes with a sign, either positive or negative. By convention, a positive chargeis a charge with the same sign as that of a proton, and a negative charge is a charge with the same sign as thatof an electron.

26 CHAPTER 3. SIGN (MATHEMATICS)

Electric charge may be positive or negative.

3.4 See also• Signedness

• Positive element

• Symmetry in mathematics

3.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 27

3.5 Text and image sources, contributors, and licenses

3.5.1 Text• Absolute value Source: https://en.wikipedia.org/wiki/Absolute_value?oldid=694241172 Contributors: AxelBoldt, Bryan Derksen, Zun-

dark, Tarquin, Andre Engels, Edemaine, Heron, Edward, Patrick, JohnOwens, Ixfd64, Dcljr, TakuyaMurata, Cgs, Rossami, Andres, Ever-cat, Pizza Puzzle, Schneelocke, Hashar, Revolver, Dcoetzee, LennyWikipedia~enwiki, Dysprosia, JitseNiesen, Furrykef, Itai, Moriel~enwiki,Rogper~enwiki, Chuunen Baka, Robbot, Rfc1394, Henrygb, Robinh, Mfc, Giftlite, Gene Ward Smith, BenFrantzDale, Ævar ArnfjörðBjarmason, Lethe, Guanaco, Yekrats, Jason Quinn, Eequor, Macrakis, OTB, Andycjp, DocSigma, LucasVB, Antandrus, Histrion, Hk-pawn~enwiki, Ukexpat, Karl Dickman, Qef, Mernen, Rich Farmbrough, Guanabot, Inkypaws, MeltBanana, Ivan Bajlo, Paul August,Syp, Rgdboer, Art LaPella, EmilJ, Spoon!, .:Ajvol:., Nk, Alansohn, ABCD, Caesura, Sleigh, Drbreznjev, InBalance, Oleg Alexandrov,Blumpkin, Rocastelo, Matijap, MFH, The wub, VKokielov, RobertG, Glenn L, Chobot, DVdm, YurikBot, Laurentius, Dmharvey, KSmrq,CambridgeBayWeather, NawlinWiki, Gwaihir, Vanished user 1029384756, Jimmyre, Bota47, Jessemerriman, Mike92591, Igiffin, Zzu-uzz, ArielGold, Gesslein, That Guy, From That Show!, SmackBot, RDBury, Incnis Mrsi, InverseHypercube, Pgk, BiT, Diegotorque-mada, Aksi great, Betacommand, ERcheck, Jweimar, Silly rabbit, Adpete, DHN-bot~enwiki, Emorgasm, Wen D House, Cybercobra,Nakon, Metebelis, LN2, Breno, Minna Sora no Shita, Jim.belk, A.Z., IronGargoyle, Ckatz, KJS77, Iridescent, Madmath789, Splitpea-soup, The editor1, AlainD, CRGreathouse, CmdrObot, FilipeS, Doctormatt, Karimarie, Gogo Dodo, He Who Is, Khattab01~enwiki,FastLizard4, Xantharius, Epbr123, LeeG, Headbomb, Marek69, Futurebird, Mentifisto, AntiVandalBot, Edokter, GiM, Hannes Eder,Dylan Lake, Rower2000, Acroterion, Magioladitis, Bongwarrior, VoABot II, Brunoman1990, Cic, David Eppstein, Gwern, FisherQueen,MartinBot, Ortensia, Highegg, Bracodbk, Avatar09, Chrishy man, J.delanoy, Defilerc, Spowage, Jacksonwalters, JonMcLoone, Policron,Aephoenix, Mviergujerghs89fhsdifds, Jamesofur, Mathsexpressions, Useight, JohnDoe0007, Idioma-bot, VolkovBot, Aesopos, PhilipTrueman, TXiKiBoT, Oshwah, Ann Stouter, Anonymous Dissident, JhsBot, Cremepuff222, Nibios, Dmcq, AlleborgoBot, SieBot, Yin-tan, JabbaTheBot, JackSchmidt, OKBot, M2Ys4U, Escape Orbit, Dlrohrer2003, Atif.t2, ClueBot, Brewcrewer, Coriakin~enwiki, Ex-cirial, Polly, Calor, 7, BlueDevil, Kiensvay, Spitfire, ZooFari, Thatguyflint, Fgnievinski, Ronhjones, WMdeMuynck, With goodness inmind, The world deserves the truth, AndersBot, SpBot, Ginosbot, Tide rolls, Zorrobot, LuK3, Legobot, Luckas-bot, Yobot, THENWHOWAS PHONE?, AnomieBOT, Jim1138, Piano non troppo, Yachtsman1, JohnnyB256, Jxramos, ArthurBot, DannyAsher, Hanberke,Isheden, Inferno, Lord of Penguins, Jhbdel, Amaury, VoItorb, FrescoBot, Majopius, Hamtechperson, Serols, SpaceFlight89, Yaddie,White Shadows, TobeBot, General Helper, Ammodramus, Aniten21, RjwilmsiBot, Q6913, Midhart90, EcneicsFlogCitanaf, Salvio giu-liano, EmausBot, Frankjohnli, Tommy2010, Wikipelli, K6ka, Slawekb, Robirahman, AvicBot, John Cline, Traxs7, C0rtesf, Usability,D.Lazard, Wayne Slam, Mcmatter, Tolly4bolly, Cit helper, TyA, Theonefoster, Vladimirdx, Chewings72, Nickminaj15, ResearchRave,ClueBot NG, KlappCK, Priyankastar, Mesoderm, O.Koslowski, Joel B. Lewis, Savantas83, Helpful Pixie Bot, Drummerboy5324, JohnCummings, Achowat, Davidfreesefan23, Rockhand, Tutelary, Henri.vanliempt, ChrisGualtieri, Travelpleb, Webclient101, Wiki2487,Conner11108, Conner111088, Jareknh, Drake Lehto, Evagavilan, BethNaught, Whikie, Wigglewigglewiggle3, SoSivr, Luisrafael1221,Milesoc, Sweepy, Nyein Myat Thu and Anonymous: 277

• Norm (mathematics) Source: https://en.wikipedia.org/wiki/Norm_(mathematics)?oldid=692392302Contributors: Zundark, TheAnome,Tomo, Patrick, Michael Hardy, SebastianHelm, Selket, Zero0000, Robbot, Altenmann, MathMartin, Bkell, Tobias Bergemann, Tosha,Connelly, Giftlite, BenFrantzDale, Lethe, Fropuff, Sendhil, Dratman, Jason Quinn, Tomruen, Almit39, Urhixidur, Beau~enwiki, Photo-Box, Sperling, Paul August, Bender235,MisterSheik, EmilJ, Dalf, Bobo192, Army1987, Bestian~enwiki, HasharBot~enwiki, Ncik~enwiki,ABCD, Oleg Alexandrov, Linas, MFH, Nahabedere, Tlroche, HannsEwald, Mike Segal, Magidin, Mathbot, ChongDae, Jenny Harrison,Tardis, Kri, CiaPan, Chobot, Algebraist, Wavelength, Eraserhead1, Hairy Dude, KSmrq, JosephSilverman, VikC, Trovatore, Vanisheduser 1029384756, Crasshopper, David Pal, Tribaal, Fmccown, Arthur Rubin, TomJF, Killerandy, Lunch, That Guy, From That Show!,SmackBot, David Kernow, InverseHypercube, Melchoir, Mhss, Bluebot, Oli Filth, Silly rabbit, Nbarth, Sbharris, Tamfang, Cícero, Cy-bercobra, DMacks, Lambiam, Dicklyon, SimonD, CBM, Irritate, MaxEnt, Mct mht, Rudjek, A876, Xtv, Thijs!bot, D4g0thur, Head-bomb, Steve Kroon, Urdutext, Selvik, Heysan, JAnDbot, Magioladitis, Reminiscenza, Chutzpan, Sullivan.t.j, ANONYMOUS COW-ARD0xC0DE, JoergenB, Robin S, Allispaul, Pharaoh of the Wizards, Lucaswilkins, Singularitarian, Potatoswatter, Idioma-bot, Cer-berus0, JohnBlackburne, PMajer, Don Quixote de la Mancha, Falcongl, Wikimorphism, Wikiisawesome, Synthebot, Free0willy, DanPolansky, RatnimSnave, Paolo.dL, MiNombreDeGuerra, JackSchmidt, ClueBot, Veromies, Baldphil, Mpd1989, Rockfang, Brews ohare,Hans Adler, Phantom xxiii, Jaan Vajakas, Addbot, Saavek47, Zorrobot, ,سعی Luckas-bot, Yobot, TaBOT-zerem, Kan8eDie, Ziyuang,SvartMan, Citation bot, Jxramos, ArthurBot, DannyAsher, Bdmy, Dlazesz, Omnipaedista, RibotBOT, Shadowjams, Quartl, FrescoBot,Paine Ellsworth, Sławomir Biały, Pinethicket, Kiefer.Wolfowitz, NearSetAccount, Stpasha, RedBot, ,enwiki~אביב Dmitri666, Datahaki,JumpDiscont, Weedwhacker128, Xnn, FoxRaweln, Tom Peleg, Jowa fan, EmausBot, Helptry, Effigies, KHamsun, ZéroBot, Midas02,Quondum, Bugmenot10, PerimeterProf, Sebjlan, Petrb, ClueBot NG, Wcherowi, Lovasoa, Snotbot, Helpful Pixie Bot, Rheyik, Aisteco,Deltahedron, Mgkrupa, Laiwoonsiu, Chandu8542 and Anonymous: 117

• Sign (mathematics) Source: https://en.wikipedia.org/wiki/Sign_(mathematics)?oldid=680559347Contributors: Zundark,Michael Hardy,Robbot, Gandalf61, Giftlite, Discospinster, Waldir, Incnis Mrsi, Jim.belk, Altamel, Magioladitis, Peskydan, Ohms law, Jdaloner, Cliff,Plastikspork, Drmies, Addbot, LaaknorBot, Sandrarossi, HerculeBot, Reinraum, Materialscientist, Xqbot, Jeffwang, Alphanis, Quondum,SporkBot, ClueBot NG, Wcherowi, Widr, Justincheng12345-bot, SteenthIWbot, The Anonymouse, Eyesnore, Yardimsever, YiFeiBot,Joeleoj123, Loraof and Anonymous: 18

3.5.2 Images• File:Absolute_value.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6b/Absolute_value.svg License: CC-BY-SA-3.0Contributors:

• Vectorised version of Image:Absolute_value.png Original artist:• This hand-written SVG version by Qef• File:Absolute_value_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/ba/Absolute_value_composition.

svg License: CC0 Contributors: Own work Original artist: Incnis Mrsi• File:Complex_conjugate_picture.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Complex_conjugate_picture.svgLicense: CC-BY-SA-3.0 Contributors: Vectorized version of http://ja.wikipedia.org/wiki/%E7%94%BB%E5%83%8F:Complex.pngwith some tweaks Original artist: Oleg Alexandrov

28 CHAPTER 3. SIGN (MATHEMATICS)

• File:Khoang_cach_tren_duong_thang_thuc.png Source: https://upload.wikimedia.org/wikipedia/commons/7/70/Khoang_cach_tren_duong_thang_thuc.png License: Public domain Contributors: Transferred from vi.wikipedia Original artist: Ah adodc at vi.wikipedia

• File:Number-line.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/93/Number-line.svgLicense: CC0Contributors: Ownwork Original artist: Hakunamenta

• File:PlusMinus.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c5/PlusMinus.svg License: Public domain Contribu-tors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine-readable authorprovided. MarianSigler assumed (based on copyright claims).

• File:Signum_function.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4f/Signum_function.svg License: CC-BY-SA-3.0 Contributors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine-readable author provided. Cronholm144 assumed (based on copyright claims).

• File:VFPt_dipole_electric.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/VFPt_dipole_electric.svg License: CCBY-SA 3.0 Contributors: This plot was created with VectorFieldPlot Original artist: Geek3

• File:Vector_norm_sup.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/ae/Vector_norm_sup.svg License: Public do-main Contributors: ? Original artist: ?

• File:Vector_norms.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4d/Vector_norms.svgLicense: CC-BY-SA-3.0Con-tributors: ? Original artist: ?

• File:Yhvyvczsa7.png Source: https://upload.wikimedia.org/wikipedia/commons/1/1f/Yhvyvczsa7.png License: Public domain Contrib-utors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine-readableauthor provided. Drandstrom assumed (based on copyright claims).

3.5.3 Content license• Creative Commons Attribution-Share Alike 3.0