Metric space

In mathematics,metric space isset (or "space") wheredistance between pointsdefined.

Tablecontents

1 History 2 Formal definition 3 Examples 4 Further definitionsproperties 4.1 Distance between pointssets 5 Related conceptsalternative axiom systems 6 See also

History

Maurice Fréchet introduced metric spaceshis work Sur quelques points du calcul fonctionnel1906.

Formal definition

Formally,metric space M issetpointsan associated distance function (also calledmetric) d : M × M -> R (where R issetreal numbers). For all x, y, zM, this functionrequiredsatisfyfollowing conditions:

1. d(x, y) ≥ 0
2. d(x, x) = 0
3. if   d(x, y) = 0   then   x = y     (identityindiscernibles)
4. d(x, y) = d(y, x)     (symmetry)
5. d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality).

These axioms express intuitive notions aboutconcept"distance": distances between different spotspositive anddistance between xy issame asdistance between yx. The triangle inequality means that if you go from xz directly, thatno longer than going first from xy,then from yz. In Euclidean geometry, thiseasysee. Metric spaces allow this conceptbe extended tomore abstract setting.

In metric spaces, one can talk about limitssequences;metric spacewhich every Cauchy sequence haslimitsaidbe complete.

Examples

• The trivial distance metric: d(x,y) = 0 if x = y else 1.
• The real numbers withdistance function d(x, y) = |y - x| given byabsolute value,more generally Euclidean n-space withEuclidean distance,complete metric spaces.
• More generally still, any normed vector space ismetric space by defining d(x, y) = ||y - x||. If suchspacecomplete, we call itBanach space.
• If Xsome setM ismetric space, thensetall bounded functions f : X -> M (i.e. those functions whose image isbounded subsetM) can be turned intometric space by defining d(f, g) = sup_xX d(f(x), g(x))any bounded functions fg. If Mcomplete, then this spacecomplete as well.
• If X istopological (or metric) spaceM ismetric space, thensetall bounded continuous functions from XM formsmetric space if we definemetric as above: d(f, g) = sup_xX d(f(x), g(x))any bounded continuous functions fg. If Mcomplete, then this spacecomplete as well.
• If M isconnected Riemannian manifold, then we can turn M intometric space by definingdistancetwo points asinfimum oflengths ofpaths (continuously differentiable curves) connecting them.
• If Gan undirected connected graph, thenset VverticesG can be turned intometric space by defining d(x, y)belength ofshortest path connectingvertices xy.
• If M ismetric space, we can turnset K(M)all compact subsetsM intometric space by definingHausdorff distance d(X, Y) = inf{r :every xX there existsyYd(x, y) < rfor every yY there exists an xX such that d(x, y) < r)}. In this metric, two elementscloseeach other if every elementone setclosesome element ofother set. One can show that K(M)complete if Mcomplete.

Further definitionsproperties

In any metric space M we can defineopen balls assets ofform

B(x; r) = {yM : d(x,y) < r},

where xin Mr ispositive real number, calledradius ofball. A subsetM which isunion(finitely or infinitely many) open ballscalled an open set. The complementan open setcalled closed. Every metric spaceautomaticallytopological space,topology beingsetall open sets. A topological space which can arisethis way frommetric spacecalledmetrizable space; seearticle on metrization theoremsfurther details.

Since metric spacestopological spaces, one hasnotioncontinuous function between metric spaces. Without referring totopology, this notion can also be directly defined using limitssequences; thisexplained inarticle on continuous functions.

A metric space Mcalled bounded if there exists some number r > 0 such that d(x,y) ≤ rall xyM (notbe confused"finite", which refers tonumberelements, nothow farset extends; finiteness implies boundedness, but not conversely). The space Mcalled totally bounded ifevery r > 0 there exist finitely many open ballsradius r whose union equals M. Itnot difficultsee that every totally bounded spacebounded. It can be shown thatmetric spacecompact ifonly if itcompletetotally bounded.

By restrictingmetric, any subset ofmetric space ismetric space itself. We call suchsubset complete, bounded, totally bounded or compact if it, considered asmetric space, hascorresponding property.

Metric spacesparacompact Hausdorff spaceshence normal (indeed theyperfectly normal). An important consequencethat every metric space admits partitionsunitythat every continuous real-valued function defined onclosed subset ofmetric space can be extended tocontinuous map onwhole space (Tietze extension theorem). Italso true that every real-valued Lipschitz-continuous map defined onsubset ofmetric space can be extended toLipschitz-continuous map onwhole space.

An isometry between two metric spaces (M₁, d₁)(M₂, d₂) isfunction f : M₁ → M₂ withproperty d₂(f(x), f(y)) = d₁(x, y)all x, yM₁. Isometriesnecessarily injective. We call two spaces isometrically isomorphic if there existsbijective isometry between them. In this case,two spacesessentially identical.

Every metric spaceisometrically isomorphic toclosed subsetsome normed vector space. Every complete metric spaceisometrically isomorphic toclosed subsetsome Banach space.

Distance between pointssets

If (M,d) ismetric space, S issubsetMx ispointM, we definedistance from xS as

d(x,S) = inf {d(x,s) : s ∈ S}

Then d(x, S) = 0 ifonly if x belongs toclosureS. Furthermore, we havefollowing generalization oftriangle inequality:

d(x,S) ≤ d(x,y) + d(y,S)

whichparticular shows thatmap x |-> d(x,S)continuous.

Related conceptsalternative axiom systems

The property 1 (d(x, y) ≥ 0) follows from properties 2, 45does not havebe required separately.

Some authors useextended real number lineallowdistance function dattainvalue ∞. Every such metric can be rescaled tofinite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) andtwo conceptsmetric spacetherefore equivalent as far as notionstopology (such as continuity or convergence)concerned.

A metriccalled an ultrametric ifsatisfiesfollowing stronger version oftriangle inequality:

• For all x, y, zM, d(x, z) ≤ max(d(x, y), d(y, z))

If one drops property 3, one obtains pseudometric spaces. Dropping property 4 instead, one obtains quasimetric spaces. However, losing symmetrythis case, one usually changes property 3 such that both d(x,y)=0d(y,x)=0neededxybe identified. All combinations ofabovepossible andreferredby their according names (such as quasi-pseudo-ultrametric).

See also

• topology
• contraction mapping