Complete Metric Space_2

Complete metric spaceFrom Wikipedia, the free encyclopedia

Chapter 1

Complete field

In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basicexamples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).

1.1 See also• Completion (ring theory)

• Hensel’s lemma

• Henselian ring

• Ostrowski’s theorem

2

Chapter 2

Complete metric space

“Cauchy completion” redirects here. For the use in category theory, see Karoubi envelope.

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence ofpoints in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.Intuitively, a space is complete if there are no “points missing” from it (inside or at the boundary). For instance,the set of rational numbers is not complete, because e.g. √2 is “missing” from it, even though one can construct aCauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to “fill allthe holes”, leading to the completion of a given space, as explained below.

2.1 Examples

The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is notcomplete. Consider for instance the sequence defined by x1 = 1 and xn+1 = xn

2 + 1xn

. This is a Cauchy sequenceof rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, thennecessarily x2 = 2, yet no rational number has this property. However, considered as a sequence of real numbers, itdoes converge to the irrational number √2.The open interval (0,1), again with the absolute value metric, is not complete either. The sequence defined by xn =1/n is Cauchy, but does not have a limit in the given space. However the closed interval [0,1] is complete; for examplethe given sequence does have a limit in this interval and the limit is zero.The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value)are complete, and so is Euclidean space Rn, with the usual distance metric. In contrast, infinite-dimensional normedvector spacesmay ormay not be complete; those that are complete are Banach spaces. The space C[a, b] of continuousreal-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respectto the supremum norm. However, the supremum norm does not give a norm on the space C(a, b) of continuousfunctions on (a, b), for it may contain unbounded functions. Instead, with the topology of compact convergence, C(a,b) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can beinduced by a complete translation-invariant metric.The spaceQp of p-adic numbers is complete for any prime number p. This space completesQ with the p-adic metricin the same way that R completes Q with the usual metric.If S is an arbitrary set, then the set SN of all sequences in S becomes a complete metric space if we define the distancebetween the sequences (xn) and (yn) to be 1/N, where N is the smallest index for which xN is distinct from yN, or 0if there is no such index. This space is homeomorphic to the product of a countable number of copies of the discretespace S.

3

4 CHAPTER 2. COMPLETE METRIC SPACE

2.2 Some theorems

A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, withdiameters tending to 0, has a non-empty intersection: if Fn is closed and non-empty, Fn ₊ ₁ ⊂ Fn for every n, anddiam(Fn) → 0, then there is a point x ∈ X common to all sets Fn.Every compact metric space is complete, though complete spaces need not be compact. In fact, a metric space iscompact if and only if it is complete and totally bounded. This is a generalization of the Heine–Borel theorem, whichstates that any closed and bounded subspace S of Rn is compact and therefore complete.[1]

A closed subspace of a complete space is complete.[2] Conversely, a complete subset of a metric space is closed.[3]

If X is a set and M is a complete metric space, then the set B(X, M) of all bounded functions f from X to M is acomplete metric space. Here we define the distance in B(X, M) in terms of the distance in M with the supremumnorm

d(f, g) ≡ sup {d[f(x), g(x)] : x ∈ X}

If X is a topological space and M is a complete metric space, then the set C (X, M) consisting of all continuousbounded functions f from X to M is a closed subspace of B(X, M) and hence also complete.The Baire category theorem says that every complete metric space is a Baire space. That is, the union of countablymany nowhere dense subsets of the space has empty interior.The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banachspaces.The expansion constant of a metric space is the infimum of all constants µ such that whenever the family {B(xα, rα)}intersects pairwise, the intersection

∩α

B(xα, µrα)

is nonempty. A metric space is complete if and only if its expansion constant is ≤ 2.[4]

2.3 Completion

For anymetric spaceM, one can construct a complete metric spaceM′ (which is also denoted asM), which containsMas a dense subspace. It has the following universal property: if N is any complete metric space and f is any uniformlycontinuous function from M to N, then there exists a unique uniformly continuous function f′ from M′ to N, whichextends f. The space M' is determined up to isometry by this property, and is called the completion of M.The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any twoCauchy sequences x=(xn) and y=(yn) in M, we may define their distance as

d(x, y) = limn

d (xn, yn)

(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since twodifferent Cauchy sequences may have the distance 0. But “having distance 0” is an equivalence relation on the set ofall Cauchy sequences, and the set of equivalence classes is a metric space, the completion ofM. The original space isembedded in this space via the identification of an element x ofM with the equivalence class of sequences convergingto x (i.e., the equivalence class containing the sequence with constant value x). This defines an isometry onto a densesubspace, as required. Notice, however, that this construction makes explicit use of the completeness of the realnumbers, so completion of the rational numbers needs a slightly different treatment.Cantor's construction of the real numbers is similar to the above construction; the real numbers are the completion ofthe rational numbers using the ordinary absolute value to measure distances. The additional subtlety to contend with is

2.4. TOPOLOGICALLY COMPLETE SPACES 5

that it is not logically permissible to use the completeness of the real numbers in their own construction. Nevertheless,equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown tobe a field that has the rational numbers as a subfield. This field is complete, admits a natural total ordering, and isthe unique totally ordered complete field (up to isomorphism). It is defined as the field of real numbers (see alsoConstruction of the real numbers for more details). One way to visualize this identification with the real numbers asusually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that “ought”to have a given real limit is identified with that real number. The truncations of the decimal expansion give just onechoice of Cauchy sequence in the relevant equivalence class.For a prime p, the p-adic numbers arise by completing the rational numbers with respect to a different metric.If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing theoriginal space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containingthe original space as a dense subspace.

2.4 Topologically complete spaces

Note that completeness is a property of the metric and not of the topology, meaning that a complete metric spacecan be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete buthomeomorphic to the open interval (0,1), which is not complete.In topology one considers completely metrizable spaces, spaces for which there exists at least one complete metricinducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be writtenas an intersection of countably many open subsets of some complete metric space. Since the conclusion of the Bairecategory theorem is purely topological, it applies to these spaces as well.Completely metrizable spaces are often called topologically complete. However, the latter term is somewhat arbitrarysince metric is not the most general structure on a topological space for which one can talk about completeness (seethe section Alternatives and generalizations). Indeed, some authors use the term topologically complete for a widerclass of topological spaces, the completely uniformizable spaces.[5]

A topological space homeomorphic to a separable complete metric space is called a Polish space.

2.5 Alternatives and generalizations

Since Cauchy sequences can also be defined in general topological groups, an alternative to relying on a metricstructure for defining completeness and constructing the completion of a space is to use a group structure. This is mostoften seen in the context of topological vector spaces, but requires only the existence of a continuous “subtraction”operation. In this setting, the distance between two points x and y is gauged not by a real number ε via the metric din the comparison d(x, y) < ε, but by an open neighbourhood N of 0 via subtraction in the comparison x − y ∈ N.A common generalisation of these definitions can be found in the context of a uniform space, where an entourage isa set of all pairs of points that are at no more than a particular “distance” from each other.It is also possible to replace Cauchy sequences in the definition of completeness by Cauchy nets or Cauchy filters. Ifevery Cauchy net (or equivalently every Cauchy filter) has a limit inX, thenX is called complete. One can furthermoreconstruct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most generalsituation in which Cauchy nets apply is Cauchy spaces; these too have a notion of completeness and completion justlike uniform spaces.

2.6 See also

• Knaster–Tarski theorem

• Completion (ring theory)

6 CHAPTER 2. COMPLETE METRIC SPACE

2.7 Notes[1] Introduction to Metric and Topological Spaces, Wilson A. Sutherland, ISBN 978-0-19-853161-6

[2] http://planetmath.org/encyclopedia/AClosedSubsetOfACompleteMetricSpaceIsComplete.html

[3] http://planetmath.org/encyclopedia/ACompleteSubspaceOfAMetricSpaceIsClosed.html

[4] B. Grünbaum, Some applications of expansion constants. Pacific J. Math. Volume 10, Number 1 (1960), 193–201.

[5] Kelley, Problem 6.L, p. 208

2.8 References• Kelley, John L. (1975). General Topology. Springer. ISBN 0-387-90125-6.

• Kreyszig, Erwin, Introductory functional analysis with applications (Wiley, New York, 1978). ISBN 0-471-03729-X

• Lang, Serge, “Real and Functional Analysis” ISBN 0-387-94001-4

• Meise, Reinhold; Vogt, Dietmar (1997). Introduction to functional analysis. Ramanujan, M.S. (trans.). Oxford:Clarendon Press; New York: Oxford University Press. ISBN 0-19-851485-9.

Chapter 3

Field (mathematics)

This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambigua-tion).

In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzerocommutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplica-tion. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfyingthe appropriate abelian group equations and distributive law. The most commonly used fields are the field of realnumbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, algebraicfunction fields, algebraic number fields, p-adic fields, and so forth.Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Thetheory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; amongother results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring thecircle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubilityof quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in numbertheory and algebraic geometry.As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fieldsallow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example theintegers form a ring, but 2x = 1 has no solution in integers. Also, the multiplication operation in a field is required tobe commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) iscalled a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields werecalled commutative fields.)As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following(not exhaustive) chain of class inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ uniquefactorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

3.1 Definition and illustration

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition andmultiplication (the latter excluding zero), with “compatible” being formalized by distributivity, and the caveat that theadditive and the multiplicative identities are distinct (0 ≠ 1).The most common way to formalize this is by defining a field as a set together with two operations, usually calledaddition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold (note thatsubtraction and division are defined in terms of the inverse operations of addition and multiplication):[note 1]

Closure of F under addition and multiplication For all a, b in F, both a + b and a · b are in F (or more formally,+ and · are binary operations on F).

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8 CHAPTER 3. FIELD (MATHEMATICS)

Associativity of addition and multiplication For all a, b, and c in F, the following equalities hold: a + (b + c) = (a+ b) + c and a · (b · c) = (a · b) · c.

Commutativity of addition and multiplication For all a and b in F, the following equalities hold: a + b = b + a anda · b = b · a.

Existence of additive and multiplicative identity elements There exists an element ofF, called the additive identityelement and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called themultiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. To exclude the trivial ring,the additive identity and the multiplicative identity are required to be distinct.

Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, suchthat a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 =1. (The elements a + (−b) and a · b−1 are also denoted a − b and a/b, respectively.) In other words, subtractionand division operations exist.

Distributivity of multiplication over addition For all a, b and c in F, the following equality holds: a · (b + c) = (a ·b) + (a · c).

A field is therefore an algebraic structure F, +, ·, −, −1, 0, 1 ; of type 2, 2, 1, 1, 0, 0 , consisting of two abeliangroups:

• F under +, −, and 0;

• F ∖ {0} under ·, −1, and 1, with 0 ≠ 1,

with · distributing over +.[1]

3.1.1 First example: rational numbers

A simple example of a field is the field of rational numbers, consisting of numbers which can be written as fractionsa/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply −a/b, and the multiplicativeinverse (provided that a ≠ 0) is b/a. To see the latter, note that

b

a· ab=

ba

ab= 1.

The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity

a

b·(c

d+

e

f

)

=a

b·(c

d· ff+

e

f· dd

)

=a

b·(cf

df+

ed

fd

)=

a

b· cf + ed

df

=a(cf + ed)

bdf=

acf

bdf+

aed

bdf=

ac

bd+

ae

bf

=a

b· cd+

a

b· ef,

or the law of commutativity and law of associativity.

3.2. RELATED ALGEBRAIC STRUCTURES 9

3.1.2 Second example: a field with four elements

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. Thefollowing example is a field consisting of four elements called O, I, A and B. The notation is chosen such that O playsthe role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1above). One can check that all field axioms are satisfied. For example:

A · (B + A) = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity.

The above field is called a finite field with four elements, and can be denoted F4. Field theory is concerned withunderstanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and describing its innerstructure. For example, from a glance at the multiplication table, it can be seen that any non-zero element (i.e., I, A,and B) is a power of A: A = A1, B = A2 = A · A, and finally I = A3 = A · A · A. This is not a coincidence, but ratherone of the starting points of a deeper understanding of (finite) fields.

3.1.3 Alternative axiomatizations

As with other algebraic structures, there exist alternative axiomatizations. Because of the relations between the op-erations, one can alternatively axiomatize a field by explicitly assuming that there are four binary operations (add,subtract, multiply, divide) with axioms relating these, or (by functional decomposition) in terms of two binary oper-ations (add and multiply) and two unary operations (additive inverse and multiplicative inverse), or other variants.The usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the otheroperations to be defined in terms of these basic ones, but in other contexts, such as topology and category theory, itis important to include all operations as explicitly given, rather than implicitly defined (compare topological group).This is because without further assumptions, the implicitly defined inverses may not be continuous (in topology), ormay not be able to be defined (in category theory). Defining an inverse requires that one is working with a set, not amore general object.For a very economical axiomatization of the field of real numbers, whose primitives are merely a set R with 1 ∈ R,addition, and a binary relation, "<". See Tarski’s axiomatization of the reals.

3.2 Related algebraic structures

The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence ofthe binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inversesare precisely the axioms for an abelian group. In other words, for any field, the subset of nonzero elements F \ {0},also often denoted F×, is an abelian group (F×, ·) usually called multiplicative group of the field. Likewise (F, +) isan abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set),obeying the distributivity.Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example,if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually calleddivision rings or skew fields.

3.2.1 Remarks

By elementary group theory, applied to the abelian groups (F×, ·), and (F, +), the additive inverse −a and the multi-plicative inverse a−1 are uniquely determined by a.Similar direct consequences from the field axioms include

−(a · b) = (−a) · b = a · (−b), in particular −a = (−1) · a

as well as

a · 0 = 0.

Both can be shown by replacing b or c with 0 in the distributive property.

10 CHAPTER 3. FIELD (MATHEMATICS)

3.3 History

The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability ofpolynomial equations with rational coefficients of degree five or higher.In 1857, Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axiomsof a field.[2] This construction has been frequently recalled as a contribution to the foundations of mathematics.In 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the four arithmeticoperations, the German word Körper, which means “body” or “corpus” (to suggest an organically closed entity),[3]hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), butthe term “a ring” (Zahlring) was invented by Hilbert.[4] In 1893, Eliakim Hastings Moore called the concept “field”in English.[5][6]

In 1881, Leopold Kronecker defined what he called a “domain of rationality”, which is indeed a field of polynomialsin modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.[7] In 1910, ErnstSteinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).[8]In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts likeprime field, perfect field and the transcendence degree of a field extension.Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.

3.4 Examples

3.4.1 Rationals and algebraic numbers

The field of rational numbersQ has been introduced above. A related class of fields very important in number theoryare algebraic number fields. We will first give an example, namely the field Q(ζ) consisting of numbers of the form

a + bζ

with a, b ∈ Q, where ζ is a primitive third root of unity, i.e., a complex number satisfying ζ3 = 1, ζ ≠ 1. This fieldextension can be used to prove a special case of Fermat’s last theorem, which asserts the non-existence of rationalnonzero solutions to the equation

x3 + y3 = z3.

In the language of field extensions detailed below, Q(ζ) is a field extension of degree 2. Algebraic number fields areby definition finite field extensions of Q, that is, fields containing Q having finite dimension as a Q-vector space.

3.4.2 Reals, complex numbers, and p-adic numbers

Take the real numbersR, under the usual operations of addition and multiplication. When the real numbers are giventhe usual ordering, they form a complete ordered field; it is this structure which provides the foundation for mostformal treatments of calculus.The complex numbers C consist of expressions

a + bi

where i is the imaginary unit, i.e., a (non-real) number satisfying i2 = −1. Addition and multiplication of real numbersare defined in such a way that all field axioms hold for C. For example, the distributive law enforces

(a + bi)·(c + di) = ac + bci + adi + bdi2, which equals ac−bd + (bc + ad)i.

The real numbers can be constructed by completing the rational numbers, i.e., filling the “gaps": for example √2 issuch a gap. By a formally very similar procedure, another important class of fields, the field of p-adic numbers Qp isbuilt. It is used in number theory and p-adic analysis.

3.4. EXAMPLES 11

Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinitenumbers.

3.4.3 Constructible numbers

Given 0, 1, r1 and r2, the construction yields r1·r2

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compassand straightedge. For example, it was unknown to the Greeks that it is in general impossible to trisect a given angle.Using the field notion and field theory allows these problems to be settled. To do so, the field of constructible numbersis considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from thesetwo by a finite number of construction steps using only compass and straightedge. This set, endowed with the usualaddition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers r1and r2 that have already been constructed can be done using construction at the right, based on the intercept theorem.This way, the obtained field F contains all rational numbers, but is bigger than Q, because for any f ∈ F, the squareroot of f is also a constructible number.A closely related concept is that of a Euclidean field, namely an ordered field whose positive elements are closed undersquare root. The real constructible numbers form the least Euclidean field, and the Euclidean fields are precisely theordered extensions thereof.

3.4.4 Finite fields

Main article: Finite field

Finite fields (also called Galois fields) are fields with finitely many elements. The above introductory example F4 is afield with four elements. F2 consists of two elements, 0 and 1. This is the smallest field, because by definition a fieldhas at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as XOR andAND operations, this field finds applications in computer science, especially in cryptography and coding theory.In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 (n repeated terms) equals 0. It can be shownthat the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field

12 CHAPTER 3. FIELD (MATHEMATICS)

has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, thecharacteristic is said to be zero.A basic class of finite fields are the fields Fp with p elements (p a prime number):

Fp = Z/pZ = {0, 1, ..., p − 1},

where the operations are defined by performing the operation in the set of integers Z, dividing by p and taking theremainder; see modular arithmetic. A field K of characteristic p necessarily contains Fp,[9] and therefore may beviewed as a vector space over Fp, of finite dimension if K is finite. Thus a finite field K has prime power order, i.e., Khas q = pn elements (where n > 0 is the number of elements in a basis of K over Fp). By developing more field theory,in particular the notion of the splitting field of a polynomial f over a field K, which is the smallest field containing Kand all roots of f, one can show that two finite fields with the same number of elements are isomorphic, i.e., there isa one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak ofthe finite field with q elements, usually denoted by Fq or GF(q).

3.4.5 Archimedean fields

Main article: Archimedean field

An Archimedean field is an ordered field such that for each element there exists a finite expression 1 + 1 + ··· + 1whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains noinfinitesimals; or, the field is isomorphic to a subfield of the reals. A necessary condition for an ordered field tobe complete is that it be Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimalnor a least positive rational, whence the sequence 1/2, 1/3, 1/4, …, every element of which is greater than everyinfinitesimal, has no limit. (And since every proper subfield of the reals also contains such gaps, up to isomorphismthe reals form the unique complete ordered field.)

3.4.6 Field of functions

Given a geometric object X, one can consider functions on such objects. Adding and multiplying them pointwise, i.e.,(f ⋅ g)(x) = f(x) ⋅ g(x) this leads to a field. However, for having multiplicative inverses, one has to consider partialfunctions, which, almost everywhere, are defined and have a non-zero value.If X is an algebraic variety over a field F, then the rational functions X → F form a field, the function field of X.This field consists of the functions that are defined and are the quotient of two polynomial functions outside somesubvariety. Likewise, if S is a Riemann surface, then the meromorphic functions S → C form a field. Under certaincircumstances, namely when S is compact, S can be reconstructed from this field.

3.4.7 Local and global fields

Another important distinction in the realm of fields, especially with regard to number theory, are local fields andglobal fields. Local fields are completions of global fields at a given place. For example, Q is a global field, andthe attached local fields are Qp and R (Ostrowski’s theorem). Algebraic number fields and function fields over Fqare further global fields. Studying arithmetic questions in global fields may sometimes be done by looking at thecorresponding questions locally—this technique is called local-global principle.

3.5 Some first theorems• Every finite subgroup of the multiplicative group F× is cyclic. This applies in particular to Fq×, it is cyclic oforder q − 1. In the introductory example, a generator of F4

× is the element A.

• A integral domain is a field if and only if it has no ideals except {0} and itself. Equivalently, an integral domainis a field if and only if its Krull dimension is 0.

• Isomorphism extension theorem

3.6. CONSTRUCTING FIELDS 13

3.6 Constructing fields

3.6.1 Closure operations

Assuming the axiom of choice, for every field F, there exists a field F, called the algebraic closure of F, which containsF, is algebraic over F, which means that any element x of F satisfies a polynomial equation

fnxn + fn₋₁xn−1 + ··· + f1x + f0 = 0, with coefficients fn, ..., f0 ∈ F,

and is algebraically closed, i.e., any such polynomial does have at least one solution in F. The algebraic closure isunique up to isomorphism inducing the identity on F. However, in many circumstances in mathematics, it is notappropriate to treat F as being uniquely determined by F, since the isomorphism above is not itself unique. In thesecases, one refers to such a F as an algebraic closure of F. A similar concept is the separable closure, containing allroots of separable polynomials, instead of all polynomials.For example, if F =Q, the algebraic closureQ is also called field of algebraic numbers. The field of algebraic numbersis an example of an algebraically closed field of characteristic zero; as such it satisfies the same first-order sentencesas the field of complex numbers C.In general, all algebraic closures of a field are isomorphic. However, there is in general no preferable isomorphismbetween two closures. Likewise for separable closures.

3.6.2 Subfields and field extensions

A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subsetcontaining 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations definedby restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, thecomputable numbers and the rational numbers are examples.The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A fieldextension F / E is simply a field F and a subfield E ⊂ F. Constructing such a field extension F / E can be done by“adding new elements” or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rationalfunctions, i.e., equivalence classes of expressions of the kind

p(X)

q(X),

where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This isthe simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomialsE in this case) being embedded into its field of fractions E(X) .The ring of formal power series E[[X]] is also a domain, and again the (equivalence classes of) fractions of the formp(X)/ q(X) where p and q are elements of E[[X]] form the field of fractions for E[[X]] . This field is actually the ringof Laurent series over the field E, denoted E((X)) .In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible howeverthat the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of realnumbersR. As explained above,C is an extension ofR.C can be obtained fromR by adjoining the imaginary symboli which satisfies i2 = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because inthat case, the powers of X are all distinct objects, but here, i2=−1 is actually an element of R.Another way to view this last example is to note that i is a zero of the polynomial p(X) = X2 + 1. The quotientring R[X]/(X2 +1) can be mapped onto C using the map a+ bX → a+ ib . Since the ideal (X2+1) is generated by apolynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map fromthe quotient to C is necessarily an isomorphism of rings.The above construction generalises to any irreducible polynomial in the polynomial ring E[X], i.e., a polynomial p(X)that cannot be written as a product of non-constant polynomials. The quotient ring F = E[X] / (p(X)), is again a field.Alternatively, constructing such field extensions can also be done, if a bigger container is already given. Supposegiven a field E, and a field G containing E as a subfield, for example G could be the algebraic closure of E. Let x be

14 CHAPTER 3. FIELD (MATHEMATICS)

an element of G not in E. Then there is a smallest subfield of G containing E and x, denoted F = E(x) and called fieldextension F / E generated by x in G.[10] Such extensions are also called simple extensions. Many extensions are of thistype; see the primitive element theorem. For instance, Q(i) is the subfield of C consisting of all numbers of the forma + bi where both a and b are rational numbers.One distinguishes between extensions having various qualities. For example, an extension K of a field k is calledalgebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is calledtranscendental. The aim of Galois theory is the study of algebraic extensions of a field.

3.6.3 Rings vs fields

Adding multiplicative inverses to an integral domain R yields the field of fractions of R. For example, the field offractions of the integers Z is just Q. Also, the field F(X) is the quotient field of the ring of polynomials F[X].Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximalideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomialp(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fieldsFp = Z / pZ.

3.6.4 Ultraproducts

If I is an index set, U is an ultrafilter on I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect toU is a field.For example, a non-principal ultraproduct of finite fields is a pseudo finite field; i.e., a PAC field having exactly oneextension of any degree.

3.7 Galois theory

Main article: Galois theory

Galois theory aims to study the algebraic extensions of a field by studying the symmetry in the arithmetic operations ofaddition and multiplication. The fundamental theorem of Galois theory shows that there is a strong relation betweenthe structure of the symmetry group and the set of algebraic extensions.In the case where F / E is a finite (Galois) extension, Galois theory studies the algebraic extensions of E that aresubfields of F. Such fields are called intermediate extensions. Specifically, the Galois group of F over E, denotedGal(F/E), is the group of field automorphisms of F that are trivial on E (i.e., the bijections σ : F → F that preserveaddition and multiplication and that send elements of E to themselves), and the fundamental theorem of Galois theorystates that there is a one-to-one correspondence between subgroups of Gal(F/E) and the set of intermediate extensionsof the extension F/E. The theorem, in fact, gives an explicit correspondence and further properties.To study all (separable) algebraic extensions of E at once, one must consider the absolute Galois group of E, definedas the Galois group of the separable closure, Esep, of E over E i.e., Gal(Esep/E). It is possible that the degree of thisextension is infinite (as in the case of E = Q). It is thus necessary to have a notion of Galois group for an infinitealgebraic extension. The Galois group in this case is obtained as a “limit” (specifically an inverse limit) of the Galoisgroups of the finite Galois extensions of E. In this way, it acquires a topology.[note 2] The fundamental theorem ofGalois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topologyof the Galois group, and in the case of Esep/E it states that there this a one-to-one correspondence between closedsubgroups of Gal(Esep/E) and the set of all separable algebraic extensions of E (technically, one only obtains thoseseparable algebraic extensions of E that occur as subfields of the chosen separable closure Esep, but since all separableclosures of E are isomorphic, choosing a different separable closure would give the same Galois group and thus an“equivalent” set of algebraic extensions).

3.8. GENERALIZATIONS 15

3.8 Generalizations

There are also proper classes with field structure, which are sometimes called Fields, with a capital F:

• The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are aproper class, not a set.

• The nimbers form a Field. The set of nimbers with birthday smaller than 22n , the nimbers with birthday smallerthan any infinite cardinal are all examples of fields.

In a different direction, differential fields are fields equipped with a derivation. For example, the field R(X), togetherwith the standard derivative of polynomials forms a differential field. These fields are central to differential Galoistheory. Exponential fields, meanwhile, are fields equipped with an exponential function that provides a homomor-phism between the additive and multiplicative groups within the field. The usual exponential function makes the realand complex numbers exponential fields, denoted Rₑₓ and Cₑₓ respectively.Generalizing in a more categorical direction yields the field with one element and related objects.

3.8.1 Exponentiation

One does not in general study generalizations of fields with three binary operations. The familiar addition/subtraction,multiplication/division, exponentiation/root-extraction/logarithm operations from the natural numbers to the reals,each built up in terms of iteration of the last, mean that generalizing exponentiation as a binary operation is tempting,but has generally not proven fruitful; instead, an exponential field assumes a unary exponential function from theadditive group to the multiplicative group, not a partially defined binary function. Note that the exponential operationof ab is neither associative nor commutative, nor has a unique inverse ( ±2 are both square roots of 4, for instance),unlike addition and multiplication, and further is not defined for many pairs—for example, (−1)1/2 =

√−1 does not

define a single number. These all show that even for rational numbers exponentiation is not nearly as well-behaved asaddition and multiplication, which is why one does not in general axiomatize exponentiation.

3.9 Applications

The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whosecomponents can be elements of an arbitrary field.Finite fields are used in number theory, Galois theory, cryptography, coding theory and combinatorics; and again thenotion of algebraic extension is an important tool.

3.10 See also

• Category of fields

• Glossary of field theory for more definitions in field theory.

• Heyting field

• Lefschetz principle

• Puiseux series

• Ring

• Vector space

• Vector spaces without fields

16 CHAPTER 3. FIELD (MATHEMATICS)

3.11 Notes

[1] That is, the axiom for addition only assumes a binary operation +: F ×F → F, a, b 7→ a+ b. The axiom of inverse allows oneto define a unary operation − : F → F a 7→ −a that sends an element to its negative (its additive inverse); this is not takenas given, but is implicitly defined in terms of addition as " −a is the unique b such that a+ b = 0 ", “implicitly” because itis defined in terms of solving an equation—and one then defines the binary operation of subtraction, also denoted by "−",as − : F ×F → F, a, b 7→ a− b := a+ (−b) in terms of addition and additive inverse. In the same way, one defines the binaryoperation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of“reciprocal” (multiplicative inverse).

[2] As an inverse limit of finite discrete groups, it is equipped with the profinite topology, making it a profinite topologicalgroup

3.12 References

[1] Wallace, D A R (1998) Groups, Rings, and Fields, SUMS. Springer-Verlag: 151, Th. 2.

[2] Karl Georg Christian v. Staudt, Beiträge zur Geometrie der Lage (Contributions to the Geometry of Position), volume 2(Nürnberg, (Germany): Bauer and Raspe, 1857). See: “Summen vonWürfen” (sums of throws), pp. 166-171 ; “Produckteaus Würfen” (products of throws), pp. 171-176 ; “Potenzen von Würfen” (powers of throws), pp. 176-182.

[3] Peter Gustav Lejeune Dirichlet with R. Dedekind, Vorlesungen über Zahlentheorie von P. G. Lejeune Dirichlet (Lectureson Number Theory by P.G. Lejeune Dirichlet), 2nd ed., volume 1 (Braunschweig, Germany: Friedrich Vieweg und Sohn,1871), p. 424. From page 424: “Unter einem Körper wollen wir jedes System von unendlich vielen reellen oder complexenZahlen verstehen, welches in sich so abgeschlossen und vollständig ist, dass die Addition, Subtraction, Multiplication undDivision von je zwei dieser Zahlen immer wieder eine Zahl desselben Systems hervorbringt.” (By a “field” we will understandany system of infinitely many real or complex numbers, which is so closed and complete that the addition, subtraction,multiplication, and division of any two of these numbers always again produces a number of the same system.)

[4] J J O'Connor and E F Robertson, The development of Ring Theory, September 2004.

[5] Moore, E. Hastings (1893), “A doubly-infinite system of simple groups”, Bulletin of the New York Mathematical Society 3(3): 73–78, doi:10.1090/S0002-9904-1893-00178-X, JFM 25.0198.01. From page 75: “Such a system of s marks [i.e., afinite field with s elements] we call a field of order s.”

[6] Earliest Known Uses of Some of the Words of Mathematics (F)

[7] Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra, Vieweg, JFM 50.0042.03

[8] Steinitz, Ernst (1910), “Algebraische Theorie der Körper”, Journal für die reine und angewandte Mathematik 137: 167–309, doi:10.1515/crll.1910.137.167, ISSN 0075-4102, JFM 41.0445.03

[9] Jacobson (2009), p. 213

[10] Jacobson (2009), p. 213

3.13 Sources

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2, especially Chapter 13

• Allenby, R.B.J.T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2

• Blyth, T.S.; Robertson, E. F. (1985), Groups, rings and fields: Algebra through practice, Cambridge UniversityPress. See especially Book 3 (ISBN 0-521-27288-2) and Book 6 (ISBN 0-521-27291-2).

• Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

• James Ax (1968), The elementary theory of finite fields, Ann. of Math. (2), 88, 239–271

3.14. EXTERNAL LINKS 17

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

• Field Theory Q&A

• Fields at ProvenMath definition and basic properties.

• Field at PlanetMath.org.

Chapter 4

Metric (mathematics)

An illustration comparing the taxicab metric versus the Euclidean metric on the plane: In the taxicab metric all three pictured paths(red, yellow, and blue) have the same length (12) for the same route. In the Euclidean metric, the green path has length 6

√2 ≈ 8.49

, and is the unique shortest path.

In mathematics, ametric or distance function is a function that defines a distance between each pair of elements of

18

4.1. DEFINITION 19

a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can begenerated by a metric. A topological space whose topology can be described by a metric is called metrizable.In differential geometry, the word “metric” may refer to a bilinear form that may be defined from the tangent vectorsof a differentiable manifold onto a scalar, allowing distances along curves to be determined through integration. It ismore properly termed a metric tensor.

4.1 Definition

Ametric on a set X is a function (called the distance function or simply distance)

d : X × X→ [0,∞),

where [0,∞) is the set of non-negative real numbers (because distance can't be negative so we can't use R), and forall x, y, z in X, the following conditions are satisfied:

Conditions 1 and 2 together define a positive-definite function. The first condition is implied by the others.A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality where pointscan never fall 'between' other points:

d(x, z) ≤ max(d(x, y), d(y, z))

for all x, y, z in X.A metric d on X is called intrinsic if any two points x and y in X can be joined by a curve with length arbitrarily closeto d(x, y).For sets on which an addition + : X × X→ X is defined, d is called a translation invariant metric if

d(x, y) = d(x + a, y + a)

for all x, y and a in X.

4.2 Notes

These conditions express intuitive notions about the concept of distance. For example, that the distance betweendistinct points is positive and the distance from x to y is the same as the distance from y to x. The triangle inequalitymeans that the distance from x to z via y is at least as great as from x to z directly. Euclid in his work stated that theshortest distance between two points is a line; that was the triangle inequality for his geometry.If a modification of the triangle inequality

4*. d(x, z) ≤ d(z, y) + d(y, x)

is used in the definition then property 1 follows straight from property 4*. Properties 2 and 4* give property 3 whichin turn gives property 4.

4.3 Examples

Main article: Metric space § Examples of metric spaces

20 CHAPTER 4. METRIC (MATHEMATICS)

• The discrete metric: if x = y then d(x,y) = 0. Otherwise, d(x,y) = 1.

• The Euclidean metric is translation and rotation invariant.

• The taxicab metric is translation invariant.

• More generally, any metric induced by a norm is translation invariant.

• If (pn)n∈N is a sequence of seminorms defining a (locally convex) topological vector space E, then

d(x, y) =∑∞

n=112n

pn(x−y)1+pn(x−y)

is a metric defining the same topology. (One can replace 12n by any summable sequence (an) of strictly

positive numbers.)

• Graph metric, a metric defined in terms of distances in a certain graph.

• The Hamming distance in coding theory.

• Riemannian metric, a type of metric function that is appropriate to impose on any differentiable manifold. Forany such manifold, one chooses at each point p a symmetric, positive definite, bilinear form L: T × T →ℝ onthe tangent space T at p, doing so in a smooth manner. This form determines the length of any tangent vectorv on the manifold, via the definition ||v|| = √L(v, v). Then for any differentiable path on the manifold, its lengthis defined as the integral of the length of the tangent vector to the path at any point, where the integration isdone with respect to the path parameter. Finally, to get a metric defined on any pair {x, y} of points of themanifold, one takes the infimum, over all paths from x to y, of the set of path lengths. A smooth manifoldequipped with a Riemannian metric is called a Riemannian manifold.

• The Fubini–Study metric on complex projective space. This is an example of a Riemannian metric.

• String metrics, such as Levenshtein distance and other string edit distances, define a metric over strings.

• Graph edit distance defines a distance function between graphs.

4.4 Equivalence of metrics

For a given set X, two metrics d1 and d2 are called topologically equivalent (uniformly equivalent) if the identitymapping

id: (X,d1) → (X,d2)

is a homeomorphism (uniform isomorphism).For example, if d is a metric, then min(d, 1) and d

1+d are metrics equivalent to d.See also notions of metric space equivalence.

4.5 Metrics on vector spaces

Norms on vector spaces are equivalent to certain metrics, namely homogeneous, translation-invariant ones. In otherwords, every norm determines a metric, and some metrics determine a norm.Given a normed vector space (X, ∥ · ∥) we can define a metric on X by

d(x, y) := ∥x− y∥

The metric d is said to be induced by the norm ∥ · ∥ .Conversely if a metric d on a vector space X satisfies the properties

4.6. METRICS ON MULTISETS 21

• d(x, y) = d(x+ a, y + a) (translation invariance)

• d(αx, αy) = |α|d(x, y) (homogeneity)

then we can define a norm on X by

∥x∥ := d(x, 0)

Similarly, a seminorm induces a pseudometric (see below), and a homogeneous, translation invariant pseudometricinduces a seminorm.

4.6 Metrics on multisets

We can generalize the notion of a metric from a distance between two elements to a distance between two nonemptyfinite multisets of elements. A multiset is a generalization of the notion of a set such that an element can occur morethan once. Define Z = XY if Z is the multiset consisting of the elements of the multisets X and Y , that is, ifx occurs once in X and once in Y then it occurs twice in Z . A distance function d on the set of nonempty finitemultisets is a metric[1] if

1. d(X) = 0 if all elements ofX are equal and d(X) > 0 otherwise (positive definiteness), that is, (non-negativityplus identity of indiscernibles)

2. d(X) is invariant under all permutations of X (symmetry)

3. d(XY ) ≤ d(XZ) + d(ZY ) (triangle inequality)

Note that the familiar metric between two elements results if the multiset X has two elements in 1 and 2 and themultisets X,Y, Z have one element each in 3. For instance if X consists of two occurrences of x , then d(X) = 0according to 1.A simple example is the set of all nonempty finite multisetsX of integers with d(X) = max{x : x ∈ X}−min{x :x ∈ X} . More complex examples are information distance in multisets;[1] and normalized compression distance(NCD) in multisets.[2]

4.7 Generalized metrics

There are numerous ways of relaxing the axioms ofmetrics, giving rise to various notions of generalizedmetric spaces.These generalizations can also be combined. The terminology used to describe them is not completely standardized.Most notably, in functional analysis pseudometrics often come from seminorms on vector spaces, and so it is naturalto call them “semimetrics”. This conflicts with the use of the term in topology.

4.7.1 Extended metrics

Some authors allow the distance function d to attain the value ∞, i.e. distances are non-negative numbers on theextended real number line. Such a function is called an extended metric or "∞-metric”. Every extended metric canbe transformed to a finite metric such that the metric spaces are equivalent as far as notions of topology (such ascontinuity or convergence) are concerned. This can be done using a subadditive monotonically increasing boundedfunction which is zero at zero, e.g. d′(x, y) = d(x, y) / (1 + d(x, y)) or d′′(x, y) = min(1, d(x, y)).The requirement that the metric take values in [0,∞) can even be relaxed to consider metrics with values in otherdirected sets. The reformulation of the axioms in this case leads to the construction of uniform spaces: topologicalspaces with an abstract structure enabling one to compare the local topologies of different points.

22 CHAPTER 4. METRIC (MATHEMATICS)

4.7.2 Pseudometrics

Main article: Pseudometric space

A pseudometric on X is a function d : X × X→ R which satisfies the axioms for a metric, except that instead of thesecond (identity of indiscernibles) only d(x,x)=0 for all x is required. In other words, the axioms for a pseudometricare:

1. d(x, y) ≥ 0

2. d(x, x) = 0 (but possibly d(x, y) = 0 for some distinct values x ̸= y .)

3. d(x, y) = d(y, x)

4. d(x, z) ≤ d(x, y) + d(y, z).

In some contexts, pseudometrics are referred to as semimetrics because of their relation to seminorms.

4.7.3 Quasimetrics

Occasionally, a quasimetric is defined as a function that satisfies all axioms for a metric with the possible exceptionof symmetry:[3][4]

1. d(x, y) ≥ 0 (positivity)

2. d(x, y) = 0 if and only if x = y (positive definiteness)

3. d(x, y) = d(y, x) (symmetry, dropped)

4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Quasimetrics are common in real life. For example, given a set X of mountain villages, the typical walking timesbetween elements of X form a quasimetric because travel up hill takes longer than travel down hill. Another exampleis a taxicab geometry topology having one-way streets, where a path from point A to point B comprises a different setof streets than a path from B to A. Nevertheless, this notion is rarely used in mathematics, and its name is not entirelystandardized.[5]

A quasimetric on the reals can be defined by setting

d(x, y) = x − y if x ≥ y, andd(x, y) = 1 otherwise. The 1 may be replaced by infinity or by 1 + 10(y−x) .

The topological space underlying this quasimetric space is the Sorgenfrey line. This space describes the process offiling down a metal stick: it is easy to reduce its size, but it is difficult or impossible to grow it.If d is a quasimetric on X, a metric d' on X can be formed by taking

d'(x, y) = 1⁄2(d(x, y) + d(y, x)).

4.7.4 Metametrics

In a metametric, all the axioms of a metric are satisfied except that the distance between identical points is notnecessarily zero. In other words, the axioms for a metametric are:

1. d(x, y) ≥ 0

2. d(x, y) = 0 implies x = y (but not vice versa.)

4.7. GENERALIZED METRICS 23

3. d(x, y) = d(y, x)

4. d(x, z) ≤ d(x, y) + d(y, z).

Metametrics appear in the study of Gromov hyperbolic metric spaces and their boundaries. The visual metametricon such a space satisfies d(x, x) = 0 for points x on the boundary, but otherwise d(x, x) is approximately the distancefrom x to the boundary. Metametrics were first defined by Jussi Väisälä.[6]

4.7.5 Semimetrics

A semimetric on X is a function d : X × X → R that satisfies the first three axioms, but not necessarily the triangleinequality:

1. d(x, y) ≥ 0

2. d(x, y) = 0 if and only if x = y

3. d(x, y) = d(y, x)

Some authors work with a weaker form of the triangle inequality, such as:

d(x, z) ≤ ρ (d(x, y) + d(y, z)) (ρ-relaxed triangle inequality)d(x, z) ≤ ρ max(d(x, y), d(y, z)) (ρ-inframetric inequality).

The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and the ρ-relaxedtriangle inequality implies the 2ρ-inframetric inequality. Semimetrics satisfying these equivalent conditions havesometimes been referred to as “quasimetrics”,[7] “nearmetrics”[8] or inframetrics.[9]

The ρ-inframetric inequalities were introduced to model round-trip delay times in the internet.[9] The triangle in-equality implies the 2-inframetric inequality, and the ultrametric inequality is exactly the 1-inframetric inequality.

4.7.6 Premetrics

Relaxing the last three axioms leads to the notion of a premetric, i.e. a function satisfying the following conditions:

1. d(x, y) ≥ 0

2. d(x, x) = 0

This is not a standard term. Sometimes it is used to refer to other generalizations ofmetrics such as pseudosemimetrics[10]or pseudometrics;[11] in translations of Russian books it sometimes appears as “prametric”.[12]

Any premetric gives rise to a topology as follows. For a positive real r, the “open” r-ball centred at a point p is definedas

Br(p) = { x | d(x, p) < r }.

A set is called open if for any point p in the set there is an “open” r-ball centred at p which is contained in the set.Every premetric space is a topological space, and in fact a sequential space. In general, the “open” r-balls themselvesneed not be open sets with respect to this topology. As for metrics, the distance between two sets A and B, is definedas

d(A, B) = infx∊A, y∊B d(x, y).

This defines a premetric on the power set of a premetric space. If we start with a (pseudosemi-)metric space, we geta pseudosemimetric, i.e. a symmetric premetric. Any premetric gives rise to a preclosure operator cl as follows:

cl(A) = { x | d(x, A) = 0 }.

24 CHAPTER 4. METRIC (MATHEMATICS)

4.7.7 Pseudoquasimetrics

The prefixes pseudo-, quasi- and semi- can also be combined, e.g., a pseudoquasimetric (sometimes called hemi-metric) relaxes both the indiscernibility axiom and the symmetry axiom and is simply a premetric satisfying thetriangle inequality. For pseudoquasimetric spaces the open r-balls form a basis of open sets. A very basic exampleof a pseudoquasimetric space is the set {0,1} with the premetric given by d(0,1) = 1 and d(1,0) = 0. The associatedtopological space is the Sierpiński space.Sets equippedwith an extended pseudoquasimetric were studied byWilliamLawvere as “generalizedmetric spaces”.[13][14]From a categorical point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, alongwith their corresponding nonexpansive maps, are the best behaved of the metric space categories. One can take ar-bitrary products and coproducts and form quotient objects within the given category. If one drops “extended”, onecan only take finite products and coproducts. If one drops “pseudo”, one cannot take quotients. Approach spaces area generalization of metric spaces that maintains these good categorical properties.

4.7.8 Important cases of generalized metrics

In differential geometry, one considers a metric tensor, which can be thought of as an “infinitesimal” quadratic metricfunction. This is defined as a nondegenerate symmetric bilinear form on the tangent space of a manifold with anappropriate differentiability requirement. While these are not metric functions as defined in this article, they inducewhat is called a pseudo-semimetric function by integration of its square root along a path through the manifold. Ifone imposes the positive-definiteness requirement of an inner product on the metric tensor, this restricts to the caseof a Riemannian manifold, and the path integration yields a metric.In general relativity the related concept is a metric tensor (general relativity) which expresses the structure of a pseudo-Riemannianmanifold. Though the term “metric” is used in cosmology, the fundamental idea is different because thereare non-zero null vectors in the tangent space of these manifolds. This generalized view of “metrics”, in which zerodistance does not imply identity, has crept into some mathematical writing too:[15][16]

4.8 See also• Acoustic metric

• Complete metric

• Similarity measure

• String metric

4.9 Notes[1] P.M.B. Vitanyi, Information distance in multiples, IEEE Trans. Inform. Theory, 57:4(2011), 2451-2456

[2] A.R.Cohen and P.M.B. Vitanyi, Normalized compression distance of multisets with applications, arXiv:1212.5711

[3] E.g. Steen & Seebach (1995).

[4] Smyth, M. (1987). M.Main, A.Melton, M.Mislove and D.Schmidt, ed. Quasi uniformities: reconciling domains with metricspaces. 3rd Conference on Mathematical Foundations of Programming Language Semantics. Springer-Verlag, LectureNotes in Computer Science 298. pp. 236–253.

[5] Rolewicz, Stefan (1987), Functional Analysis and Control Theory: Linear Systems, Springer, ISBN 90-277-2186-6, OCLC13064804 This book calls them “semimetrics”. That same term is also frequently used for two other generalizations ofmetrics.

[6] Väisälä, Jussi (2005), “Gromov hyperbolic spaces” (PDF),ExpositionesMathematicae 23 (3): 187–231, doi:10.1016/j.exmath.2005.01.010,MR 2164775

[7] Xia, Q. (2009), “TheGeodesic Problem inQuasimetric Spaces”, Journal ofGeometric Analysis 19 (2): 452–479, doi:10.1007/s12220-008-9065-4

4.10. REFERENCES 25

[8] Qinglan Xia (2008), “The geodesic problem in nearmetric spaces”, Journal of Geometric Analysis: Volume , Issue (009),Page 19 (2): 452–479, arXiv:0807.3377.

[9] • Fraigniaud, P.; Lebhar, E.; Viennot, L. (2008), “2008 IEEE INFOCOM - The 27th Conference on Computer Com-munications”, IEEE INFOCOM2008. the 27th Conference on Computer Communications: 1085–1093, doi:10.1109/INFOCOM.2008.163,ISBN 978-1-4244-2026-1, retrieved 2009-04-17 |chapter= ignored (help).

[10] Buldygin, V.V.; Kozachenko, I.U.V. (2000), Metric characterization of random variables and random processes.

[11] Khelemskiĭ (2006), Lectures and exercises on functional analysis.

[12] Arkhangel’skii & Pontryagin (1990). Aldrovandi, R.; Pereira, J.G. (1995), An introduction to geometrical physics.

[13] Lawvere, F.W. (2002) [1973],Metric spaces, generalised logic, and closed categories, Reprints in Theory and Applicationsof Categories 1, pp. 1–37.

[14] Vickers, Steven (2005), “Localic completion of generalized metric spaces I”, Theory and Applications of Categories 14:328–356

[15] S. Parrott (1987) Relativistic Electrodynamics and Differential Geometry, page 4, Springer-Verlag ISBN 0-387-96435-5 :“This bilinear form is variously called the Lorentz metric, or Minkowski metric or metric tensor.”

[16] Thomas E. Cecil (1992) Lie Sphere Geometry, page 9, Springer-Verlag ISBN 0-387-97747-3 : “We call this scalar productthe Lorentz metric"

4.10 References• Arkhangel’skii, A. V.; Pontryagin, L. S. (1990), General Topology I: Basic Concepts and Constructions Dimen-sion Theory, Encyclopaedia of Mathematical Sciences, Springer, ISBN 3-540-18178-4

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology, Dover, ISBN 978-0-486-68735-3, MR 507446, OCLC 32311847

4.11 External links• Quasimetric space at PlanetMath.org.

• Semimetric at PlanetMath.org.

Chapter 5

Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined. Thosedistances, taken together, are called a metric on the set. A metric on a space induces topological properties like openand closed sets, which lead to the study of more abstract topological spaces.The most familiar metric space is 3-dimensional Euclidean space. In fact, a “metric” is the generalization of theEuclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metricdefines the distance between two points as the length of the straight line segment connecting them. Other metricspaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured byangle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space ofvelocities.

5.1 History

Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat.Palermo 22 (1906) 1–74.

5.2 Definition

Ametric space is an ordered pair (M,d) whereM is a set and d is a metric onM , i.e., a function

d : M ×M → R

such that for any x, y, z ∈ M , the following holds:[1]

The first condition follows from the other three. Since for any x, y ∈ M :

The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for ametric space if it is clear from the context what metric is used.Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be definedas the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads.The triangle inequality expresses the fact that detours aren't shortcuts. Many of the examples below can be seen asconcrete versions of this general idea.

26

5.3. EXAMPLES OF METRIC SPACES 27

5.3 Examples of metric spaces• The real numbers with the distance function d(x, y) = |y − x| given by the absolute difference, and moregenerally Euclidean n -space with the Euclidean distance, are complete metric spaces. The rational numberswith the same distance also form a metric space, but are not complete.

• The positive real numbers with distance function d(x, y) = | log(y/x)| is a complete metric space.

• Any normed vector space is a metric space by defining d(x, y) = ∥y − x∥ , see also metrics on vector spaces.(If such a space is complete, we call it a Banach space.) Examples:

• The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points,or vectors, is the sum of the differences between corresponding coordinates.

• The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal numberof moves a chess king would take to travel from x to y .

• The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space isgiven by d(x, y) = ∥x∥ + ∥y∥ for distinct points x and y , and d(x, x) = 0 . More generally ∥.∥ can bereplaced with a function f taking an arbitrary set S to non-negative reals and taking the value 0 at most once:then the metric is defined on S by d(x, y) = f(x) + f(y) for distinct points x and y , and d(x, x) = 0 . Thename alludes to the tendency of railway journeys (or letters) to proceed via London (or Paris) irrespective oftheir final destination.

• If (M,d) is a metric space and X is a subset of M , then (X, d) becomes a metric space by restricting thedomain of d to X ×X .

• The discrete metric, where d(x, y) = 0 if x = y and d(x, y) = 1 otherwise, is a simple but importantexample, and can be applied to all sets. This, in particular, shows that for any set, there is always a metric spaceassociated to it. Using this metric, any point is an open ball, and therefore every subset is open and the spacehas the discrete topology.

• A finite metric space is a metric space having a finite number of points. Not every finite metric space can beisometrically embedded in a Euclidean space.[2][3]

• The hyperbolic plane is a metric space. More generally:

• IfM is any connected Riemannian manifold, then we can turnM into a metric space by defining the distanceof two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.

• If X is some set and M is a metric space, then, the set of all bounded functions f : X → M (i.e. thosefunctions whose image is a bounded subset of M ) can be turned into a metric space by defining d(f, g) =supx∈X d(f(x), g(x)) for any two bounded functions f and g (where sup is supremum).[4] This metric is calledthe uniform metric or supremum metric, and If M is complete, then this function space is complete as well.If X is also a topological space, then the set of all bounded continuous functions fromX toM (endowed withthe uniform metric), will also be a complete metric if M is.

• If G is an undirected connected graph, then the set V of vertices of G can be turned into a metric space bydefining d(x, y) to be the length of the shortest path connecting the vertices x and y . In geometric grouptheory this is applied to the Cayley graph of a group, yielding the word metric.

• Graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graphedit operations required to transform one graph into another.

• The Levenshtein distance is a measure of the dissimilarity between two strings u and v , defined as the minimalnumber of character deletions, insertions, or substitutions required to transform u into v . This can be thoughtof as a special case of the shortest path metric in a graph and is one example of an edit distance.

• Given a metric space (X, d) and an increasing concave function f : [0,∞) → [0,∞) such that f(x) = 0 ifand only if x = 0 , then f ◦ d is also a metric on X .

• Given an injective function f from any set A to a metric space (X, d) , d(f(x), f(y)) defines a metric on A .

• Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several typesof analysis.

28 CHAPTER 5. METRIC SPACE

• The set of allm by n matrices over some field is a metric space with respect to the rank distance d(X,Y ) =rank(Y −X) .

• The Helly metric is used in game theory.

5.4 Open and closed sets, topology and convergence

Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about generaltopological spaces also apply to all metric spaces.About any point x in a metric spaceM we define the open ball of radius r > 0 (where r is a real number) aboutx as the set

B(x; r) = {y ∈ M : d(x, y) < r}.

These open balls form the base for a topology on M, making it a topological space.Explicitly, a subset U ofM is called open if for every x in U there exists an r > 0 such that B(x; r) is contained inU . The complement of an open set is called closed. A neighborhood of the point x is any subset ofM that containsan open ball about x as a subset.A topological space which can arise in this way from a metric space is called a metrizable space; see the article onmetrization theorems for further details.A sequence ( xn ) in a metric spaceM is said to converge to the limit x ∈ M iff for every ϵ > 0 , there exists a naturalnumber N such that d(xn, x) < ϵ for all n > N . Equivalently, one can use the general definition of convergenceavailable in all topological spaces.A subset A of the metric spaceM is closed iff every sequence in A that converges to a limit inM has its limit in A .

5.5 Types of metric spaces

5.5.1 Complete spaces

Main article: Complete metric space

Ametric spaceM is said to be complete if every Cauchy sequence converges inM . That is to say: if d(xn, xm) → 0as both n andm independently go to infinity, then there is some y ∈ M with d(xn, y) → 0 .Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using theabsolute value metric d(x, y) = |x− y| , are not complete.Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given spaceas a dense subset. For example, the real numbers are the completion of the rationals.IfX is a complete subset of the metric spaceM , thenX is closed inM . Indeed, a space is complete iff it is closedin any containing metric space.Every complete metric space is a Baire space.

5.5.2 Bounded and totally bounded spaces

See also: bounded set

A metric space M is called bounded if there exists some number r, such that d(x,y) ≤ r for all x and y in M. Thesmallest possible such r is called the diameter of M. The space M is called precompact or totally bounded if forevery r > 0 there exist finitely many open balls of radius r whose union coversM. Since the set of the centres of theseballs is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded

5.5. TYPES OF METRIC SPACES 29

A

diam(A)

Diameter of a set.

space is bounded. The converse does not hold, since any infinite set can be given the discrete metric (one of theexamples above) under which it is bounded and yet not totally bounded.Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space Rn

a bounded set is referred to as “a finite interval” or “finite region”. However boundedness should not in general beconfused with “finite”, which refers to the number of elements, not to how far the set extends; finiteness impliesboundedness, but not conversely. Also note that an unbounded subset of Rn may have a finite volume.

5.5.3 Compact spaces

Ametric spaceM is compact if every sequence inM has a subsequence that converges to a point inM. This is knownas sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological

30 CHAPTER 5. METRIC SPACE

notions of countable compactness and compactness defined via open covers.Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaceswith finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact.A metric space is compact iff it is complete and totally bounded. This is known as the Heine–Borel theorem. Notethat compactness depends only on the topology, while boundedness depends on the metric.Lebesgue’s number lemma states that for every open cover of a compact metric space M, there exists a “Lebesguenumber” δ such that every subset of M of diameter < δ is contained in some member of the cover.Every compact metric space is second countable,[5] and is a continuous image of the Cantor set. (The latter result isdue to Pavel Alexandrov and Urysohn.)

5.5.4 Locally compact and proper spaces

A metric space is said to be locally compact if every point has a compact neighborhood. Euclidean spaces are locallycompact, but infinite-dimensional Banach spaces are not.A space is proper if every closed ball {y : d(x,y) ≤ r} is compact. Proper spaces are locally compact, but the converseis not true in general.

5.5.5 Connectedness

A metric spaceM is connected if the only subsets that are both open and closed are the empty set andM itself.A metric space M is path connected if for any two points x, y ∈ M there exists a continuous map f : [0, 1] → Mwith f(0) = x and f(1) = y . Every path connected space is connected, but the converse is not true in general.There are also local versions of these definitions: locally connected spaces and locally path connected spaces.Simply connected spaces are those that, in a certain sense, do not have “holes”.

5.5.6 Separable spaces

A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers orany Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to secondcountability and also to the Lindelöf property.

5.6 Types of maps between metric spaces

Suppose (M1,d1) and (M2,d2) are two metric spaces.

5.6.1 Continuous maps

Main article: Continuous function (topology)

The map f:M1→M2 is continuous if it has one (and therefore all) of the following equivalent properties:

General topological continuity for every open set U in M2, the preimage f −1(U) is open in M1

This is the general definition of continuity in topology.

Sequential continuity if (xn) is a sequence in M1 that converges to x in M1, then the sequence (f(xn)) convergesto f(x) in M2.

This is sequential continuity, due to Eduard Heine.

5.6. TYPES OF MAPS BETWEEN METRIC SPACES 31

ε-δ definition for every x in M1 and every ε>0 there exists δ>0 such that for all y in M1 we have

d1(x, y) < δ ⇒ d2(f(x), f(y)) < ε.

This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover, f is continuous if and only if it is continuous on every compact subset of M1.The image of every compact set under a continuous function is compact, and the image of every connected set undera continuous function is connected.

5.6.2 Uniformly continuous maps

The map ƒ : M1 → M2 is uniformly continuous if for every ε > 0 there exists δ > 0 such that

d1(x, y) < δ ⇒ d2(f(x), f(y)) < ε for all x, y ∈ M1.

Every uniformly continuous map ƒ : M1 →M2 is continuous. The converse is true ifM1 is compact (Heine–Cantortheorem).Uniformly continuous maps turn Cauchy sequences in M1 into Cauchy sequences in M2. For continuous maps thisis generally wrong; for example, a continuous map from the open interval (0,1) onto the real line turns some Cauchysequences into unbounded sequences.

5.6.3 Lipschitz-continuous maps and contractions

Given a number K > 0, the map ƒ : M1 → M2 is K-Lipschitz continuous if

d2(f(x), f(y)) ≤ Kd1(x, y) for all x, y ∈ M1.

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.If K < 1, then ƒ is called a contraction. SupposeM2 =M1 andM1 is complete. If ƒ is a contraction, then ƒ admits aunique fixed point (Banach fixed point theorem). If M1 is compact, the condition can be weakened a bit: ƒ admits aunique fixed point if

d(f(x), f(y)) < d(x, y) for all x ̸= y ∈ M1

5.6.4 Isometries

The map f:M1→M2 is an isometry if

d2(f(x), f(y)) = d1(x, y) for all x, y ∈ M1

Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete,respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed(or open).

5.6.5 Quasi-isometries

The map f : M1 → M2 is a quasi-isometry if there exist constants A ≥ 1 and B ≥ 0 such that

32 CHAPTER 5. METRIC SPACE

1

Ad2(f(x), f(y))−B ≤ d1(x, y) ≤ Ad2(f(x), f(y)) +B all for x, y ∈ M1

and a constant C ≥ 0 such that every point in M2 has a distance at most C from some point in the image f(M1).Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the “large-scale structure” ofmetric spaces; they find use in geometric group theory in relation to the word metric.

5.7 Notions of metric space equivalence

Given two metric spaces (M1, d1) and (M2, d2):

• They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism between them(i.e., a bijection continuous in both directions).

• They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e.,a bijection uniformly continuous in both directions).

• They are called isometric if there exists a bijective isometry between them. In this case, the two metric spacesare essentially identical.

• They are called quasi-isometric if there exists a quasi-isometry between them.

5.8 Topological properties

Metric spaces are paracompact[6] Hausdorff spaces[7] and hence normal (indeed they are perfectly normal). Animportant consequence is that every metric space admits partitions of unity and that every continuous real-valuedfunction defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietzeextension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metricspace can be extended to a Lipschitz-continuous map on the whole space.Metric spaces are first countable since one can use balls with rational radius as a neighborhood base.The metric topology on a metric spaceM is the coarsest topology onM relative to which the metric d is a continuousmap from the product of M with itself to the non-negative real numbers.

5.9 Distance between points and sets; Hausdorff distance and Gromovmetric

A simple way to construct a function separating a point from a closed set (as required for a completely regular space)is to consider the distance between the point and the set. If (M,d) is a metric space, S is a subset ofM and x is a pointof M, we define the distance from x to S as

d(x, S) = inf{d(x, s) : s ∈ S} where inf represents the infimum.

Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of thetriangle inequality:

d(x, S) ≤ d(x, y) + d(y, S),

which in particular shows that the map x 7→ d(x, S) is continuous.Given two subsets S and T of M, we define their Hausdorff distance to be

5.10. PRODUCT METRIC SPACES 33

dH(S, T ) = max{sup{d(s, T ) : s ∈ S}, sup{d(t, S) : t ∈ T}} where sup represents the supremum.

In general, the Hausdorff distance dH(S,T) can be infinite. Two sets are close to each other in the Hausdorff distanceif every element of either set is close to some element of the other set.The Hausdorff distance dH turns the set K(M) of all non-empty compact subsets of M into a metric space. One canshow that K(M) is complete ifM is complete. (A different notion of convergence of compact subsets is given by theKuratowski convergence.)One can then define the Gromov–Hausdorff distance between any two metric spaces by considering the minimalHausdorff distance of isometrically embedded versions of the two spaces. Using this distance, the class of all (isometryclasses of) compact metric spaces becomes a metric space in its own right.

5.10 Product metric spaces

If (M1, d1), . . . , (Mn, dn) aremetric spaces, andN is the Euclidean norm onRn, then(M1×. . .×Mn, N(d1, . . . , dn)

)is a metric space, where the product metric is defined by

N(d1, ..., dn)((x1, . . . , xn), (y1, . . . , yn)

)= N

(d1(x1, y1), . . . , dn(xn, yn)

),

and the induced topology agrees with the product topology. By the equivalence of norms in finite dimensions, anequivalent metric is obtained if N is the taxicab norm, a p-norm, the max norm, or any other norm which is non-decreasing as the coordinates of a positive n-tuple increase (yielding the triangle inequality).Similarly, a countable product of metric spaces can be obtained using the following metric

d(x, y) =

∞∑i=1

1

2idi(xi, yi)

1 + di(xi, yi).

An uncountable product of metric spaces need not be metrizable. For example, RR is not first-countable and thusisn't metrizable.

5.10.1 Continuity of distance

It is worth noting that in the case of a single space (M,d) , the distance map d : M ×M → R+ (from the definition)is uniformly continuous with respect to any of the above product metrics N(d, d) , and in particular is continuouswith respect to the product topology ofM ×M .

5.11 Quotient metric spaces

If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define

d′([x], [y]) = inf{d(p1, q1) + d(p2, q2) + · · ·+ d(pn, qn)}

where the infimum is taken over all finite sequences (p1, p2, . . . , pn) and (q1, q2, . . . , qn) with [p1] = [x] , [qn] = [y], [qi] = [pi+1], i = 1, 2, . . . , n − 1 . In general this will only define a pseudometric, i.e. d′([x], [y]) = 0 doesnot necessarily imply that [x] = [y] . However, for nice equivalence relations (e.g., those given by gluing togetherpolyhedra along faces), it is a metric.The quotient metric d is characterized by the following universal property. If f : (M,d) −→ (X, δ) is a metric mapbetween metric spaces (that is, δ(f(x), f(y)) ≤ d(x, y) for all x, y) satisfying f(x)=f(y) whenever x ∼ y, then theinduced function f : M/ ∼−→ X , given by f([x]) = f(x) , is a metric map f : (M/ ∼, d′) −→ (X, δ).

A topological space is sequential if and only if it is a quotient of a metric space.[8]

34 CHAPTER 5. METRIC SPACE

5.12 Generalizations of metric spaces

• Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topologicalspace. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.

• If we consider the first definition of a metric space given above and relax the second requirement, we arriveat the concepts of a pseudometric space or a dislocated metric space.[9] If we remove the third or fourth, wearrive at a quasimetric space, or a semimetric space.

• If the distance function takes values in the extended real number line R∪{+∞}, but otherwise satisfies all fourconditions, then it is called an extended metric and the corresponding space is called an ∞ -metric space. If thedistance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly),then we arrive at the notion of generalized ultrametric.[10]

• Approach spaces are a generalization ofmetric spaces, based on point-to-set distances, instead of point-to-pointdistances.

• A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions ofmetric spaces and domains.

• A partial metric space is intended to be the least generalisation of the notion of a metric space, such that thedistance of each point from itself is no longer necessarily zero.[11]

5.12.1 Metric spaces as enriched categories

The ordered set (R,≥) can be seen as a category by requesting exactly one morphism a → b if a ≥ b and noneotherwise. By using+ as the tensor product and 0 as the identity, it becomes a monoidal category R∗ . Every metricspace (M,d) can now be viewed as a categoryM∗ enriched over R∗ :

• Set Ob(M∗) := M

• For each X,Y ∈ M set Hom(X,Y ) := d(X,Y ) ∈ Ob(R∗)

• The composition morphism Hom(Y, Z) ⊗ Hom(X,Y ) → Hom(X,Z) will be the unique morphism in R∗

given from the triangle inequality d(y, z) + d(x, y) ≥ d(x, z)

• The identity morphism 0 → Hom(X,X) will be the unique morphism given from the fact that 0 ≥ d(X,X) .

• Since R∗ is a poset, all diagrams that are required for an enriched category commute automatically.

See the paper by F.W. Lawvere listed below.

5.13 See also

• Space (mathematics)

• Metric (mathematics)

• Metric signature

• Metric tensor

• Metric tree

• Norm (mathematics)

• Normed vector space

5.14. NOTES 35

• Measure (mathematics)

• Hilbert space

• Product metric

• Aleksandrov–Rassias problem

• Category of metric spaces

• Classical Wiener space

• Glossary of Riemannian and metric geometry

• Isometry, contraction mapping and metric map

• Lipschitz continuity

• Triangle inequality

5.14 Notes[1] B. Choudhary (1992). The Elements of Complex Analysis. New Age International. p. 20. ISBN 978-81-224-0399-2.

[2] Nathan Linial. Finite Metric Spaces—Combinatorics, Geometry and Algorithms, Proceedings of the ICM, Beijing 2002,vol. 3, pp573–586

[3] Open problems on embeddings of finite metric spaces, edited by Jirīı Matoušek, 2007

[4] Searcóid, p. 107.

[5] PlanetMath: a compact metric space is second countable

[6] Rudin, Mary Ellen. A new proof that metric spaces are paracompact. Proceedings of the American Mathematical Society,Vol. 20, No. 2. (Feb., 1969), p. 603.

[7] metric spaces are Hausdorff at PlanetMath.org.

[8] Goreham, Anthony. Sequential convergence in Topological Spaces. Honours’ Dissertation, Queen’s College, Oxford (April,2001), p. 14

[9] Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC,2010.

[10] Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC,2010.

[11] http://www.dcs.warwick.ac.uk/pmetric/

5.15 References• Victor Bryant, Metric Spaces: Iteration and Application, Cambridge University Press, 1985, ISBN 0-521-31897-1.

• Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society,2001, ISBN 0-8218-2129-6.

• Athanase Papadopoulos,Metric Spaces, Convexity and Nonpositive Curvature, European Mathematical Society,First edition 2004, ISBN 978-3-03719-010-4. Second edition 2014, ISBN 978-3-03719-132-3.

• Mícheál Ó Searcóid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006, ISBN 1-84628-369-8.

• Lawvere, F. William, “Metric spaces, generalized logic, and closed categories”, [Rend. Sem. Mat. Fis. Milano43 (1973), 135—166 (1974); (Italian summary)

36 CHAPTER 5. METRIC SPACE

This is reprinted (with author commentary) at Reprints in Theory and Applications of Categories Also (with an authorcommentary) in Enriched categories in the logic of geometry and analysis. Repr. Theory Appl. Categ. No. 1 (2002),1–37.

• Weisstein, Eric W., “Product Metric”, MathWorld.

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

• Far and near — several examples of distance functions at cut-the-knot.

Chapter 6

Real number

For the real numbers used in descriptive set theory, see Baire space (set theory). For the computing datatype, seeFloating-point number.In mathematics, a real number is a value that represents a quantity along a continuous line. The adjective real in

A symbol of the set of real numbers (ℝ)

37

38 CHAPTER 6. REAL NUMBER

this context was introduced in the 17th century by Descartes, who distinguished between real and imaginary roots ofpolynomials.The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrationalnumbers, such as √2 (1.41421356…, the square root of 2, an irrational algebraic number). Included within theirrationals are the transcendental numbers, such as π (3.14159265…, a transcendental number). Real numbers canbe thought of as points on an infinitely long line called the number line or real line, where the points corresponding tointegers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, suchas that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The realline can be thought of as a part of the complex plane, and complex numbers include real numbers.

Real numbers can be thought of as points on an infinitely long number line

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics.The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition wasneeded – was one of themost important developments of 19th centurymathematics. The currently standard axiomaticdefinition is that real numbers form the unique complete totally ordered field (R ; + ; · ; <), up to an isomorphism,[1]whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchysequences of rational numbers, Dedekind cuts, or certain infinite “decimal representations”, together with preciseinterpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm ofclassical mathematics.The reals are uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinitesets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set ofall real numbers (denoted c and called cardinality of the continuum) is strictly greater than the cardinality of the setof all natural numbers (denoted ℵ0 ). The statement that there is no subset of the reals with cardinality strictly greaterthan ℵ0 and strictly smaller than c is known as the continuum hypothesis (CH). It is known to be neither provable norrefutable using the axioms of Zermelo–Fraenkel set theory (ZFC), the standard foundation of modern mathematics,in the sense that some models of ZFC satisfy CH, while others violate it.

6.1 History

Simple fractions have been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" (“The rules of chords”)in, c. 600 BC, include what may be the first “use” of irrational numbers. The concept of irrationality was implicitlyaccepted by early Indian mathematicians since Manava (c. 750–690 BC), who were aware that the square roots ofcertain numbers such as 2 and 61 could not be exactly determined.[2] Around 500 BC, the Greek mathematicians ledby Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2.The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first by Indian andChinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbersas algebraic objects,[3] which was made possible by the development of algebra. Arabic mathematicians merged theconcepts of "number" and "magnitude" into a more general idea of real numbers.[4] The Egyptian mathematician AbūKāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations oras coefficients in an equation, often in the form of square roots, cube roots and fourth roots.[5]

In the 16th century, Simon Stevin created the basis formodern decimal notation, and insisted that there is no differencebetween rational and irrational numbers in this regard.

6.2. DEFINITION 39

In the 17th century, Descartes introduced the term “real” to describe roots of a polynomial, distinguishing them from“imaginary” ones.In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Johann HeinrichLambert (1761) gave the first flawed proof that π cannot be rational; Adrien-Marie Legendre (1794) completed theproof,[6] and showed that π is not the square root of a rational number.[7] Paolo Ruffini (1799) and Niels Henrik Abel(1842) both constructed proofs of the Abel–Ruffini theorem: that the general quintic or higher equations cannot besolved by a general formula involving only arithmetical operations and roots.Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radi-cals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e2 can be aroot of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor(1873) extended and greatly simplified this proof.[8] Charles Hermite (1873) first proved that e is transcendental,and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann’s proof was much simplified byWeierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitzand Paul Gordan.The development of calculus in the 18th century used the entire set of real numbers without having defined themcleanly. The first rigorous definition was given by Georg Cantor in 1871. In 1874, he showed that the set of allreal numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widelyheld beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor’s firstuncountability proof.

6.2 Definition

Main article: Construction of the real numbers

The real number system (R; +; ·;<) can be defined axiomatically up to an isomorphism, which is described hereafter.There are also many ways to construct “the” real number system, for example, starting from natural numbers, thendefining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchysequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start fromsome rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number systemgeometrically. From the structuralist point of view all these constructions are on equal footing.

6.2.1 Axiomatic approach

Let ℝ denote the set of all real numbers. Then:

• The set ℝ is a field, meaning that addition and multiplication are defined and have the usual properties.

• The field ℝ is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:

• if x ≥ y then x + z ≥ y + z;• if x ≥ 0 and y ≥ 0 then xy ≥ 0.

• The order is Dedekind-complete; that is: every non-empty subset S of ℝ with an upper bound in ℝ has a leastupper bound (also called supremum) in ℝ.

The last property is what differentiates the reals from the rationals. For example, the set of rationals with square lessthan 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is notrational.The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-completeordered fields ℝ1 and ℝ2, there exists a unique field isomorphism from ℝ1 to ℝ2, allowing us to think of them asessentially the same mathematical object.For another axiomatization of ℝ, see Tarski’s axiomatization of the reals.

40 CHAPTER 6. REAL NUMBER

6.2.2 Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence definedby a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; …) converges to a unique real number, in thiscase π. For details and other constructions of real numbers, see construction of the real numbers.

6.3 Properties

6.3.1 Basic properties

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, orzero. Real numbers are used to measure continuous quantities. They may be expressed by decimal representationsthat have an infinite sequence of digits to the right of the decimal point; these are often represented in the same formas 324.823122147… The ellipsis (three dots) indicates that there would still be more digits to come.More formally, real numbers have the two basic properties of being an ordered field, and having the least upper boundproperty. The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound. Thesecond condition distinguishes the real numbers from the rational numbers: for example, the set of rational numberswhose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence therational numbers do not satisfy the least upper bound property.

6.3.2 Completeness

Main article: Completeness of the real numbers

A main reason for using real numbers is that the reals contain all limits. More precisely, every sequence of realnumbers having the property that consecutive terms of the sequence become arbitrarily close to each other necessarilyhas the property that after some term in the sequence the remaining terms are arbitrarily close to some specific realnumber. In mathematical terminology, this means that the reals are complete (in the sense of metric spaces oruniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). Thisis formally defined in the following way:A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possiblydepending on ε) such that the distance |xn − xm| is less than ε for all n and m that are both greater than N. In otherwords, a sequence is a Cauchy sequence if its elements xn eventually come and remain arbitrarily close to each other.A sequence (xn) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) suchthat the distance |xn − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if itselements eventually come and remain arbitrarily close to x.Notice that every convergent sequence is a Cauchy sequence. The converse is also true:

Every Cauchy sequence of real numbers is convergent to a real number.

That is: the reals are complete.Note that the rationals are not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421…), whereeach term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not convergeto a rational number. (In the real numbers, in contrast, it converges to the positive square root of 2.)The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standardnumerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically notknown in advance.For example, the standard series of the exponential function

6.3. PROPERTIES 41

ex =∞∑

n=0

xn

n!

converges to a real number because for every x the sums

M∑n=N

xn

n!

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we knowthat the sequence converges even if the limit is not known in advance.

6.3.3 “The complete ordered field”

The real numbers are often described as “the complete ordered field”, a phrase that can be interpreted in several ways.First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it canhave no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.Additionally, an order can be Dedekind-complete, as defined in the sectionAxioms. The uniqueness result at the endof that section justifies using the word “the” in the phrase “complete ordered field” when this is the sense of “complete”that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts,since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in astandard way.These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additivegroup of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology);the description in the previous section Completeness is a special case. (We refer to the notion of completenessin uniform spaces rather than the related and better known notion for metric spaces, since the definition of metricspace relies on already having a characterization of the real numbers.) It is not true that R is the only uniformlycomplete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears thephrase “complete Archimedean field” instead of “complete ordered field”. Every uniformly complete Archimedeanfield must also be Dedekind-complete (and vice versa, of course), justifying using “the” in the phrase “the completeArchimedean field”. This sense of completeness is most closely related to the construction of the reals from Cauchysequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals)and forms the uniform completion of it in a standard way.But the original use of the phrase “complete Archimedean field” was by David Hilbert, who meant still something elseby it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedeanfield is a subfield of R. Thus R is “complete” in the sense that nothing further can be added to it without making it nolonger an Archimedean field. This sense of completeness is most closely related to the construction of the reals fromsurreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) andthen selects from it the largest Archimedean subfield.

6.3.4 Advanced properties

See also: Real line

The reals are uncountable; that is: there are strictly more real numbers than natural numbers, even though both setsare infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the naturalnumbers, and Cantor’s diagonal argument states that the latter set’s cardinality is strictly greater than the cardinalityof N. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existenceof a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuumhypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms ofset theory.As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, isdense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountableand have the same cardinality as the reals.

42 CHAPTER 6. REAL NUMBER

The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. Byvirtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and theone arising from the order are identical, but yield different presentations for the topology – in the order topologyas ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topol-ogy presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals are acontractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them;for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.Every nonnegative real number has a square root in R, although no negative number does. This shows that the orderon R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root:these two properties make R the premier example of a real closed field. Proving this is the first half of one proof ofthe fundamental theorem of algebra.The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as atopological group normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers thatare not Lebesgue measurable, e.g. Vitali sets.The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It isnot possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that thereexists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as thereal numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fieldsthat satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandardanalysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it inR), we know that the same statement must also be true of R.The field R of real numbers is an extension field of the field Q of rational numbers, and R can therefore be seen asa vector space over Q. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis ofthis vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finitelinear combination of elements of this set, using rational coefficients only, and such that no element of B is a rationallinear combination of the others. However, this existence theorem is purely theoretical, as such a base has never beenexplicitly described.The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed:there exists a total order on R with the property that every non-empty subset of R has a least element in this ordering.(The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain aleast element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not beenexplicitly described. If V=L is assumed in addition to the axioms of ZF, a well ordering of the real numbers can beshown to be explicitly definable by a formula.[9]

6.4 Applications and connections to other areas

6.4.1 Real numbers and logic

The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but somemathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbersare also studied in reverse mathematics and in constructive mathematics.[10]

The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the realnumbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closerto the original intuitions of Leibniz, Euler, Cauchy and others.Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unarypredicate “standard”. In this approach, infinitesimals are (non-"standard”) elements of the set of the real numbers(rather than being elements of an extension thereof, as in Robinson’s theory).The continuum hypothesis posits that the cardinality of the set of the real numbers is ℵ1 ; i.e. the smallest infinitecardinal number after ℵ0 , the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independentof the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiomof set theory, without contradiction.

6.5. VOCABULARY AND NOTATION 43

6.4.2 In physics

In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables,such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physicaltheories such as classical mechanics, electromagnetism, quantummechanics, general relativity and the standard modelare described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the realnumbers, although actual measurements of physical quantities are of finite accuracy and precision.In some recent developments of theoretical physics stemming from the holographic principle, the Universe is seenfundamentally as an information store, essentially zeroes and ones, organized in much less geometrical fashion andmanifesting itself as space-time and particle fields only on a more superficial level. This approach removes the realnumber system from its foundational role in physics and even prohibits the existence of infinite precision real numbersin the physical universe by considerations based on the Bekenstein bound.[11]

6.4.3 In computation

With some exceptions, most calculators do not operate on real numbers. Instead, they work with finite-precisionapproximations called floating-point numbers. In fact, most scientific computation uses floating-point arithmetic.Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not.Computers cannot directly store arbitrary real numbers with infinitely many digits.The precision is limited by the number of bits allocated to store a number, whether as floating-point numbers orarbitrary precision numbers. However, computer algebra systems can operate on irrational quantities exactly bymanipulating formulas for them (such as

√2 , arcsin

(223

), or

∫ 1

0xx dx ) rather than their rational or decimal

approximation;[12] however, it is not in general possible to determine whether two such expressions are equal (theconstant problem).A real number is called computable if there exists an algorithm that yields its digits. Because there are only countablymany algorithms,[13] but an uncountable number of reals, almost all real numbers fail to be computable. Moreover,the equality of two computable numbers is an undecidable problem. Some constructivists accept the existence ofonly those reals that are computable. The set of definable numbers is broader, but still only countable.

6.4.4 “Reals” in set theory

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since thelatter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire spaceare referred to as “reals”.

6.5 Vocabulary and notation

Mathematicians use the symbol R, or, alternatively, ℝ, the letter “R” in blackboard bold (encoded in Unicode asU+211Dℝ double-struck capital r (HTML&#8477;)), to represent the set of all real numbers. As this set is naturallyendowed with the structure of a field, the expression field of real numbers is frequently used when its algebraicproperties are under consideration.The sets of positive real numbers and negative real numbers are often noted R+ and R−,[14] respectively; R₊ and R₋are also used.[15] The non-negative real numbers can be noted R≥₀ but one often sees this set noted R+ ∪ {0}.[14] InFrench mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets arenoted respectively ℝ₊ and ℝ₋.[15] In this understanding, the respective sets without zero are called strictly positive realnumbers and strictly negative real numbers, and are noted ℝ₊* and ℝ₋*.[15]

The notationRn refers to the cartesian product of n copies ofR, which is an n-dimensional vector space over the fieldof the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon asa coordinate system has been chosen in the latter. For example, a value from R3 consists of three real numbers andspecifies the coordinates of a point in 3‑dimensional space.In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or thereal field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning

44 CHAPTER 6. REAL NUMBER

a real number (as in “the set of all reals”).

6.6 Generalizations and extensions

The real numbers can be generalized and extended in several different directions:

• The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed fieldunlike the real numbers. However, the complex numbers are not an ordered field.

• The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longera field, not even an additive group, but it still has a total order; moreover, it is a complete lattice.

• The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, noteven an additive group. However, it allows division of a non-zero element by zero. It has cyclic order describedby a separation relation.

• The long real line pastes together ℵ1* + ℵ1 copies of the real line plus a single point (here ℵ1* denotes thereversed ordering of ℵ1) to create an ordered set that is “locally” identical to the real numbers, but somehowlonger; for instance, there is an order-preserving embedding of ℵ1 in the long real line but not in the realnumbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with theprevious two examples, this set is no longer a field or additive group.

• Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them containinfinitesimal and infinitely large numbers and are therefore non-Archimedean ordered fields.

• Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize thereals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvaluesare real and they form a real associative algebra. Positive-definite operators correspond to the positive realsand normal operators correspond to the complex numbers.

6.7 See also• Complex number

• Continued fraction

• Hypercomplex number

• Imaginary number

• Limit of a sequence

• Natural number

• Real analysis

6.8 Notes[1] More precisely, given two complete totally ordered fields, there is a unique isomorphism between them. This implies that

the identity is the unique field automorphism of the reals that is compatible with the ordering.

[2] T. K. Puttaswamy, “TheAccomplishments of Ancient IndianMathematicians”, pp. 410–1. In: Selin, Helaine; D'Ambrosio,Ubiratan, eds. (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2.

[3] O'Connor, John J.; Robertson, Edmund F., “Arabic mathematics: forgotten brilliance?",MacTutor History of Mathematicsarchive, University of St Andrews.

[4] Matvievskaya, Galina (1987), “The Theory of Quadratic Irrationals in Medieval Oriental Mathematics”, Annals of the NewYork Academy of Sciences 500: 253–277 [254], doi:10.1111/j.1749-6632.1987.tb37206.x

6.9. REFERENCES 45

[5] Jacques Sesiano, “Islamic mathematics”, p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics AcrossCultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2

[6] Beckmann, Petr (1993),AHistory of Pi, Dorset Classic Reprints, Barnes&Noble Publishing, p. 170, ISBN9780880294188.

[7] Arndt, Jörg; Haenel, Christoph (2001), Pi Unleashed, Springer, p. 192, ISBN 9783540665724.

[8] Dunham, William (2015), The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press, p.127, ISBN 9781400866793, Cantor found a remarkable shortcut to reach Liouville’s conclusion with a fraction of thework

[9] Moschovakis, Yiannis N. Descriptive set theory. Studies in Logic and the Foundations ofMathematics, 100. North-HollandPublishing Co., Amsterdam - New York, 1980. xii+637 pp. ISBN 0-444-85305-7. Chapter V.

[10] Bishop, Errett; Bridges, Douglas (1985), Constructive analysis, Grundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Sciences] 279, Berlin, New York: Springer-Verlag, ISBN 978-3-540-15066-4, chapter2.

[11] Scott Aaronson, NP-complete Problems and Physical Reality, ACM SIGACT News, vol. 36, no. 1. (March 2005), pp.30–52.

[12] Cohen, Joel S. (2002), Computer algebra and symbolic computation: elementary algorithms 1, A K Peters, p. 32, ISBN978-1-56881-158-1

[13] James L. Hein, Discrete Structures, Logic, and Computability, 3rd edition (Jones and Bartlett Publishers, Sudbury, Mas-sachusetts, USA), section 14.1.1 (2010).

[14] Schumacher 1996, pp. 114-115

[15] École Normale Supérieure of Paris, “Nombres réels” (“Real numbers”), p. 6

6.9 References• Georg Cantor, 1874, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal fürdie Reine und Angewandte Mathematik, volume 77, pages 258–262.

• Solomon Feferman, 1989, The Number Systems: Foundations of Algebra and Analysis, AMS Chelsea, ISBN0-8218-2915-7.

• Robert Katz, 1964, Axiomatic Analysis, D. C. Heath and Company.

• Edmund Landau, 2001, ISBN 0-8218-2693-X, Foundations of Analysis, American Mathematical Society.

• Howie, John M., Real Analysis, Springer, 2005, ISBN 1-85233-314-6.

• Schumacher, Carol (1996), ChapterZero / Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4.

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

• The real numbers: Pythagoras to Stevin

• The real numbers: Stevin to Hilbert

• The real numbers: Attempts to understand

• What are the “real numbers,” really?

46 CHAPTER 6. REAL NUMBER

6.11 Text and image sources, contributors, and licenses

6.11.1 Text• Complete field Source: https://en.wikipedia.org/wiki/Complete_field?oldid=545546751 Contributors: RobHar, JackSchmidt, Addbot,

Obersachsebot, Ringspectrum and Qetuth• Completemetric space Source: https://en.wikipedia.org/wiki/Complete_metric_space?oldid=694118695Contributors: AxelBoldt, LC~enwiki,

Zundark, Toby Bartels, Miguel~enwiki, Patrick, Michael Hardy, TakuyaMurata, Dori, Snoyes, Jonathan Chang, Hashar, Jitse Niesen,AndrewKepert, SirJective, Aenar, Robbot, Romanm, MathMartin, Robinh, Tobias Bergemann, Tosha, Giftlite, Lethe, Vasile, Hellisp,Avihu, Noisy, Rich Farmbrough, Guanabot, Sligocki, Oleg Alexandrov, Isnow, Salix alba, Mike Segal, Eubot, Chobot, Gwaihir, Trova-tore, DYLAN LENNON~enwiki, Weppens, That Guy, From That Show!, SmackBot, Incnis Mrsi, Zeycus, Bluebot, Kurykh, Silly rabbit,Tekhnofiend, Sct72, Danpovey, Juan Daniel López, Khattab01~enwiki, Thijs!bot, BehnamFarid, Dugwiki, Salgueiro~enwiki, JAnDbot,Magioladitis, Sullivan.t.j, Patstuart, VolkovBot, A4bot, Picojeff, Spinningspark, Stca74, BotMultichill, Messagetolove, Msrasnw, Cheese-fondue, Chaley67, Jaan Vajakas, Loewepeter, Oboylej10, SoSaysChappy, CarsracBot, Legobot, Yobot, Erel Segal, Bdmy, Druiffic, The-helpfulbot, Tcnuk, Trappist the monk, EmausBot, Chharvey, Zephyrus Tavvier, MerlIwBot, Helpful Pixie Bot, Beaumont877, Sboosali,Dexbot and Anonymous: 51

• Field (mathematics) Source: https://en.wikipedia.org/wiki/Field_(mathematics)?oldid=701908590Contributors: AxelBoldt, BryanDerk-sen, Zundark, The Anome, Andre Engels, Josh Grosse, XJaM, Toby Bartels, Miguel~enwiki, Lir, Patrick, Michael Hardy, Wshun,DIG~enwiki, TakuyaMurata, Karada, Looxix~enwiki, Rossami, Andres, Loren Rosen, Revolver, RodC, Dysprosia, Jitse Niesen, Prumpf,Tero~enwiki, Phys, Philopp, R3m0t, Jmabel, Mattblack82, MathMartin, P0lyglut, Wikibot, Tobias Bergemann, Unfree, Marc Venot,Giftlite, Highlandwolf, Gene Ward Smith, Lethe, Zigger, Fropuff, Millerc, Waltpohl, Python eggs, Gubbubu, CSTAR, Pmanderson,Barnaby dawson, PhotoBox, Mormegil, Jørgen Friis Bak, Discospinster, Guanabot, Sperling, Paul August, Zaslav, Elwikipedista~enwiki,El C, Rgdboer, EmilJ, Touriste, Army1987, Giraffedata, Obradovic Goran, OoberMick, Msh210, Mlm42, Olegalexandrov, RJFJR, OlegAlexandrov, Woohookitty, Linas, Arneth, Bkkbrad, Hypercube~enwiki, MarkTempeit, Damicatz, MFH, Isnow, Palica, Graham87, Fre-plySpang, Chenxlee, Josh Parris, Rjwilmsi, Hiberniantears, Salix alba, R.e.b., FlaBot, Codazzi~enwiki, Jrtayloriv, R160K, Chobot, AbuAmaal, Algebraist, Wavelength, Dmesg, Eraserhead1, Hairy Dude, KSmrq, Grubber, Archelon, Rintrah, Rat144, Rick Norwood, Trova-tore, DYLAN LENNON~enwiki, Crasshopper, RaSten, DavidHouse~enwiki, Mgnbar, Children of the dragon, SmackBot, Mmernex,Melchoir, Gilliam, Nbarth, Charlotte Hobbs, Lesnail, Cybercobra, Acepectif, Slawekk, Bidabadi~enwiki, Lambiam, Jim.belk, Schildt.a,Mets501, DabMachine, Rschwieb, WAREL, Newone, Vaughan Pratt, CRGreathouse, Kupirijo, Tiphareth, DEWEY, Eulerianpath, Pe-dro Fonini, Goldencako, BobNiichel, Xantharius, KLIP~enwiki, JLISP, Headbomb, RobHar, Nick Number, Turgidson, Kprateek88,Martinkunev, Magioladitis, Bongwarrior, VoABot II, JamesBWatson, Jakob.scholbach, SwiftBot, Catgut, Lukeaw, MORI, Cpiral, Map-room, Gombang, Policron, Barylior, Umarekawari, LokiClock, Red Act, Anonymous Dissident, Hesam7, Joeldl, Dave703, Zermalo,Shellgirl, Cwkmail, Soler97, JackSchmidt, Jorgen W, Anchor Link Bot, Willy, your mate, Oekaki, UKe-CH, ClueBot, JP.Martin-Flatin,Mild Bill Hiccup, Tcklein, Niceguyedc, He7d3r, Bender2k14, Squirreljml, Palnot, ZooFari, Addbot, Gabriele ricci, Download, Unz-erlegbarkeit, Cesiumfrog, Yobot, Ht686rg90, TaBOT-zerem, Zagothal, AnomieBOT, UBJ 43X, DSisyphBot, Depassp, Danielschreiber,MegaMouthBolt123, Point-set topologist, Charvest, KirarinSnow, FrescoBot, Mjmarkowitz, RandomDSdevel, Ebony Jackson, D stankov,Girish.ponkiya2007, Kunle102, DASHBot, Sedrikov, Tom.kemp90, Tommy2010,Wikipelli, Shishir332, Lfrazier11, Quondum, D.Lazard,JimMeiss, ClueBot NG, Ankur1vi, Wcherowi, Frietjes, MerlIwBot, Helpful Pixie Bot, !mcbloobyenstein!!, Or elharar, Fabio.nsantos,Rjs.swarnkar, Topgraph28, Deltahedron, Sanipriya, GigaGerard, CsDix, YiFeiBot, UY Scuti, Teddyktchan, GeoffreyT2000, WillemienH,Charlotte Aryanne, Vluczkow, Gameravsgaos and Anonymous: 134

• Metric (mathematics) Source: https://en.wikipedia.org/wiki/Metric_(mathematics)?oldid=702569107Contributors: XJaM,Heron, Tomo,Patrick, Michael Hardy, Dominus, Kku, TakuyaMurata, Charles Matthews, Jitse Niesen, Dosei, Altenmann, MathMartin, JerryFriedman,Tobias Bergemann, Tosha, Giftlite, BenFrantzDale, Lethe, Jason Quinn, Nielmo~enwiki, PhotoBox, TedPavlic, El C, Rgdboer, ObradovicGoran, Crust, Apatterno, Caesura, Masatran, R., Oleg Alexandrov, Linas, WadeSimMiser, MFH, Arikan~enwiki, Qwertyus, MarSch,Salix alba, VKokielov, Mathbot, DVdm, Adoniscik, Algebraist, YurikBot, Hairy Dude, Trovatore, Multichill, Crasshopper, E Wing,Pred,Mosher, SmackBot, Reedy,Mhss, Kurykh, Nbarth, Jhausauer, Daqu, Loadmaster, Jergosh, TooMuchMath,Mets501, JMK,MOBle,Roland.barrat, HenningThielemann, Farzaneh, Kilva, Rlupsa, JustAGal, Valandil211, AntiVandalBot, Lovibond, Salgueiro~enwiki, DavidEppstein, Robin S, Pere prlpz, Peskydan, DarwinPeacock, JohnBlackburne,Marcosaedro, Guswen, Flyer22Reborn,MiNombreDeGuerra,Hans Adler, Aitias, Addbot, Okcash, Yobot, ,1971إماراتي Ark12~enwiki, AnomieBOT, Viennot, Citation bot, Twri, Dougabug, Bdmy,Weichaoliu, Ct529, Fortdj33, Citation bot 1, DrilBot, Pinethicket, I dream of horses, Trappist the monk, Alvaro Vidal-Abarca, Wis-api, GoingBatty, Weleepoxypoo, Vthierry, Usability, Quondum, SporkBot, ClueBot NG, ~enwiki, Helpful Pixie Bot, BG19bot,Catrincm, Brirush, Mark viking, Behroozomidvar, Catrinski, David9550, Carsten.edits.wiki, Ganatuiyop, Scicurious, Biomolecular-Graphics4All and Anonymous: 66

• Metric space Source: https://en.wikipedia.org/wiki/Metric_space?oldid=702571998 Contributors: AxelBoldt, LC~enwiki, Zundark,Tarquin, XJaM, Toby Bartels, Edemaine, Paul Ebermann, Tomo, Patrick, Chas zzz brown, Michael Hardy, Wshun, SGBailey, Takuya-Murata, Looxix~enwiki, Andres, Tristanb, Ideyal, Revolver, RodC, CharlesMatthews, Dcoetzee, Dfeuer, Dysprosia, Jitse Niesen, Prumpf,Saltine, AndrewKepert, Mtcv, AnanthaRaman, Donarreiskoffer, Robbot, RedWolf, Altenmann, Romanm,MathMartin, Robinh, Aetheling,Tobias Bergemann, Tosha, Giftlite, Bob Palin, Gene Ward Smith, Markus Krötzsch, Lupin, Fropuff, David Johnson, Python eggs, Gub-bubu, LiDaobing, Vivacissamamente, [email protected], Pyrop, TedPavlic, Guanabot, Paul August, SpookyMulder, BACbKA,Kinitawowi, El C, Rgdboer, Crisófilax, Miraage, Blotwell, Emhoo~enwiki, Obradovic Goran, Helix84, Tsirel, Jumbuck, Eric Kvaalen,Sligocki, Fiedorow, Themillofkeytone, Pashi, Kbolino, Oleg Alexandrov, Saeed, Joriki, Linas, Tabletop, Plowboylifestyle, Nileshbansal,Marudubshinki, Graham87, BD2412, Salix alba, Brighterorange, Mathbot, Margosbot~enwiki, Vulturejoe, Jenny Harrison, CiaPan,Chobot, YurikBot, Hairy Dude, Calumny, Number 57, Stefan Udrea, Kompik, Nothlit, JahJah, Pred, MullerHolk, RonnieBrown, Sar-danaphalus, SmackBot, Incnis Mrsi, Hammerite, PJTraill, Complexica, Nbarth, Hve, Xyzzy n, Meni Rosenfeld, Hyperwired, Mets501,Madmath789, Buckyboy314, Roland.barrat, CRGreathouse, KerryVeenstra, CmdrObot, Jackzhp, Thijs!bot, Epbr123, Kilva, Rlupsa,Seanskye, Urdutext, Orionus, Azaghal of Belegost, Dougher, JAnDbot, MER-C, Quentar~enwiki, Douglas Whitaker, Extropian314,Magioladitis, Wlod, JJ Harrison, David Eppstein, Oravec, MartinBot, TheSeven, Policron, Undernearththeman, GSpeight, LokiClock,Moswiki, TXiKiBoT, Plclark, Wikimorphism, Gregogil, Chirpstation, CenturionZ 1, Psymun747, SieBot, YonaBot, WereSpielChe-quers, Garde, Paolo.dL, MiNombreDeGuerra, Jorgen W, Skeptical scientist, Anchor Link Bot, Melcombe, Cliff, UKoch, Lbertolotti,Hans Adler, Vanished user tj23rpoij4tikkd, DumZiBoT, XLinkBot, Charles Sturm, Addbot, Tjlaxs, Haruth, LaaknorBot, Dyaa, Ok-cash, Zorrobot, Luckas-bot, Yobot, Ciphers, Joule36e5, Bdmy, Syena, Defeng.wu, DrilBot, HRoestBot, Kiefer.Wolfowitz, Sra2114,

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• Real number Source: https://en.wikipedia.org/wiki/Real_number?oldid=702284551 Contributors: Damian Yerrick, AxelBoldt, BrionVIBBER, BryanDerksen, Zundark, Tarquin, AstroNomer, Andre Engels, XJaM,Christian List, Toby~enwiki, TobyBartels, Miguel~enwiki,Stevertigo, Patrick, Michael Hardy, Wshun, Dominus, Gabbe, Chinju, TakuyaMurata, Anonymous56789, Eric119, Ejrh, Ahoerstemeier,Iulianu, Stevenj, Den fjättrade ankan~enwiki, Cyan, Andres, Panoramix, Pizza Puzzle, Ideyal, Charles Matthews, Dysprosia, Jitse Niesen,Doradus, Hyacinth, Tero~enwiki, Populus, Omegatron, Jerzy, Aleph4, Robbot, Josh Cherry, Fredrik, Benwing, R3m0t, Romanm, ChrisRoy, Markcollinsx, Stewartadcock, Rasmus Faber, Fuelbottle, Miles, Tobias Bergemann, Tosha, Giftlite, Ian Maxwell, Gene Ward Smith,Ævar Arnfjörð Bjarmason, Lethe, Herbee, Fropuff, No Guru, Curps, Varlaam, Dmmaus, Nayuki, Macrakis, Tweenk, Doshell, SlowkingMan, Lockeownzj00, Joeblakesley, Pmanderson, Elroch, SamHocevar, WpZurp, Klemen Kocjancic, PhotoBox, Mike Rosoft, Rich Farm-brough, Guanabot, Notinasnaid, Mani1, Paul August, Bender235, Jaberwocky6669, Nabla, Brian0918, El C, Rgdboer, Renfield, Bobo192,SpeedyGonsales, Nk, MPerel, Eddideigel, LutzL, Jcrocker, Rehernan~enwiki, Jumbuck, Msh210, Free Bear, Arthena, AzaToth, Balsterneb, Velella, Dirac1933, Alai, Oleg Alexandrov, CONFIQ, Linas, Isnow, Frungi, Marudubshinki, Paxsimius, Graham87, DePiep, Jsha-dias, Josh Parris, Sjakkalle, Carwil, SudoMonas, Bertik, Zbxgscqf, Salix alba, Pabix, R.e.b., The wub, VKokielov, Nowhither, KarlFrei,Jrtayloriv, Fresheneesz, Sodin, Glenn L, Chobot, Karch, DVdm, Siddhant, YurikBot, Wavelength, Borgx, X42bn6, Charles Gaudette,Dmharvey, RussBot, Taejo, Gaius Cornelius, Wimt, NawlinWiki, Grafen, Arichnad, Trovatore, Długosz, DYLAN LENNON~enwiki,IvanDurak, Elizabeyth, Bota47, Acetic Acid, Poochy, Ms2ger, Saric, Lt-wiki-bot, Omtay38, Arthur Rubin, CWenger, Gesslein, Grin-Bot~enwiki, Mejor Los Indios, Marquez~enwiki, SmackBot, RDBury, FocalPoint, Tarret, 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RedSoxstudent2, Blackbombchu, My name is not dave, W. 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48 CHAPTER 6. REAL NUMBER

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