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- ...ric space consists of two components, a set and a metric in that set. In a metric space, the metric replaces the Euclidean distance as a notion of "distance" betwe Every [[simple graph|simple]] [[graph]] can be viewed as a metric space (in more than one way). Thus formally the theory of simple graphs can be co6 KB (1,068 words) - 07:30, 4 January 2009
- 92 bytes (13 words) - 09:56, 4 September 2009
- ...in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete." Let ''X'' be a metric space with metric ''d''. Then ''X'' is complete if for every Cauchy sequence <mat3 KB (441 words) - 12:23, 4 January 2009
- 12 bytes (1 word) - 19:27, 10 November 2007
- 699 bytes (87 words) - 12:20, 4 January 2009
- Auto-populated based on [[Special:WhatLinksHere/Metric space]]. Needs checking by a human. {{r|Complete metric space}}942 bytes (125 words) - 18:29, 11 January 2010
- 12 bytes (1 word) - 12:20, 4 January 2009
- 123 bytes (18 words) - 12:20, 4 January 2009
- {{r|Metric space}}297 bytes (43 words) - 12:20, 4 January 2009
Page text matches
- A complete metric space in which a given metric space is isometrically embedded as a dense subspace.136 bytes (20 words) - 07:38, 4 January 2009
- * A discrete metric space is [[complete metric space|complete]]456 bytes (71 words) - 12:47, 4 January 2009
- * In mathematics, a function defining a [[metric space]]163 bytes (20 words) - 12:51, 31 May 2009
- Auto-populated based on [[Special:WhatLinksHere/Metric space]]. Needs checking by a human. {{r|Complete metric space}}942 bytes (125 words) - 18:29, 11 January 2010
- ...alently, the intersection of any sequence of open dense sets in a complete metric space is dense.199 bytes (30 words) - 06:18, 4 September 2009
- {{r|Complete metric space}}423 bytes (60 words) - 15:14, 28 July 2009
- {{r|Metric space}}359 bytes (48 words) - 15:04, 28 July 2009
- ...on ''X'' as <math>d(x,y)=\|x-y\|</math>. Hence a normed space is also a [[metric space]].880 bytes (157 words) - 22:28, 20 February 2010
- ...in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete." Let ''X'' be a metric space with metric ''d''. Then ''X'' is complete if for every Cauchy sequence <mat3 KB (441 words) - 12:23, 4 January 2009
- #REDIRECT [[Metric space#Mappings]]35 bytes (4 words) - 07:24, 4 January 2009
- In [[mathematics]], a '''totally bounded set''' is any [[set|subset]] of a [[metric space]] with the property that for any positive radius ''r>0'' it is contained i ...sts a finite number ''n''(''r'') (that depends on the value of ''r'') of [[metric space#Metric topology|open balls]] of radius ''r'', <math>B_r(x_1),\ldots,B_r(x_{975 bytes (166 words) - 15:27, 6 January 2009
- #REDIRECT [[Complete metric space/Bibliography]]48 bytes (5 words) - 12:20, 4 January 2009
- #REDIRECT [[Complete metric space#Topologically complete space]]64 bytes (7 words) - 12:42, 4 January 2009
- #REDIRECT [[Complete metric space/Related Articles]]52 bytes (6 words) - 12:20, 4 January 2009
- * An indiscrete space is [[metric space|metrizable]] if and only if it has at most one point766 bytes (106 words) - 16:04, 4 January 2013
- * A G<sub>δ</sub> in a [[complete metric space]] is again a complete metric space.1 KB (223 words) - 13:16, 8 March 2009
- * [[Complete metric space]]120 bytes (13 words) - 12:25, 4 January 2009
- ...[prime number]] ''p'', on the field '''Q''' of [[rational number]]s is a [[metric space|metric]] which is a [[valuation]] on the field. The ''p''-adic metric on '''Q''' is not [[complete metric space|complete]]: the [[p-adic number]]s are the corresponding [[completion]].1 KB (168 words) - 12:39, 4 January 2009
- The Cantor set is a [[complete metric space]] with respect to ''d''.2 KB (306 words) - 16:51, 31 January 2011
- {{r|Metric space}}297 bytes (43 words) - 12:20, 4 January 2009
- The real numbers form a [[metric space]] with the usual distance as metric. As a [[topological space]], a subset A Euclidean space of fixed finite dimension ''n'' also forms a [[metric space]] with the Euclidean distance as metric. As a [[topological space]], the s2 KB (381 words) - 08:54, 29 December 2008
- ...em that states the existence and uniqueness of a fixed-point in a complete metric space.130 bytes (19 words) - 05:07, 16 January 2012
- {{r|Metric space}}531 bytes (72 words) - 14:37, 31 October 2008
- {{r|Metric space}}565 bytes (76 words) - 19:05, 11 January 2010
- In [[mathematics]], a '''Cauchy sequence''' is a [[sequence]] in a [[metric space]] with the property that elements in that sequence ''cluster'' together mor ...ences may be convergent or not. This leads to the notion of a ''[[complete metric space]]'' as one in which every Cauchy sequence converges to a point of the space1 KB (240 words) - 12:30, 4 January 2009
- The extended non-negative real exponent associated to any metric space where the Hausdorff measure changes from ∞ to 0.158 bytes (20 words) - 07:55, 16 January 2012
- ...other forms throughout mathematics, and is encountered in the theory of [[metric space]]s in topology, the theory of [[normed vector space]]s in functional analys ...uality is ''assumed'' as one of the axioms for a metric space. Formally, a metric space is a set <math>X</math> equipped with a distance function <math>d: X \times2 KB (414 words) - 08:12, 16 April 2009
- ...ric space consists of two components, a set and a metric in that set. In a metric space, the metric replaces the Euclidean distance as a notion of "distance" betwe Every [[simple graph|simple]] [[graph]] can be viewed as a metric space (in more than one way). Thus formally the theory of simple graphs can be co6 KB (1,068 words) - 07:30, 4 January 2009
- ...''. The ''[[Baire category theorem]]'' states that a non-empty [[complete metric space]] is of second category.850 bytes (118 words) - 22:30, 20 February 2010
- A subset of a metric space with the property that for any positive radius it is coveted by a finite un172 bytes (30 words) - 11:56, 28 December 2008
- {{r|Metric space}}322 bytes (45 words) - 13:51, 26 July 2008
- {{r|Complete metric space}}566 bytes (74 words) - 16:25, 11 January 2010
- ...ch known as [[functional analysis]], a '''Hilbert space''' is a [[complete metric space|complete]] [[inner product space]]. As such, it is automatically also a [[B2 KB (258 words) - 12:33, 4 January 2009
- {{r|Metric space}}576 bytes (77 words) - 19:04, 11 January 2010
- As a mathematical term, '''geometry''' refers to either the spatial ([[metric space|metric]]) properties of a given space or, more specifically in [[differenti2 KB (232 words) - 03:09, 8 March 2024
- ===Metric space=== In a [[metric space]] (''X'',''d''), a limit point of a set ''S'' may be defined as a point ''x2 KB (385 words) - 22:53, 17 February 2009
- ...Baire category theorem''' states that a non-[[empty set|empty]] [[complete metric space]] is a [[second category space]]: that is, it is not a [[countability|count501 bytes (67 words) - 23:00, 5 February 2009
- {{r|Complete metric space}}689 bytes (88 words) - 17:15, 11 January 2010
- {{r|Metric space}}1 KB (169 words) - 19:54, 11 January 2010
- ===Function on a metric space=== A function ''f'' from a [[metric space]] <math>(X,d)</math> to another metric space <math>(Y,e)</math> is ''continuous'' at a point <math>x_0 \in X</math> if f3 KB (614 words) - 14:20, 13 November 2008
- *[[Complete metric space]]389 bytes (39 words) - 12:37, 4 January 2009
- {{r|Metric space}}518 bytes (68 words) - 18:06, 11 January 2010
- {{r|Complete metric space}}940 bytes (149 words) - 15:13, 28 July 2009
- {{r|Complete metric space}}955 bytes (150 words) - 15:15, 28 July 2009
- {{r|Complete metric space}}241 bytes (34 words) - 12:31, 4 January 2009
- ...math>\|x\|=\langle x,x \rangle^{1/2}</math>. Therefore it also induces a [[metric space#metric topology|metric topology]] on ''X'' via the metric <math>d(x,y)=\|x-3 KB (511 words) - 00:25, 20 February 2010
- {{r|Metric space}}959 bytes (152 words) - 15:06, 28 July 2009
- In a [[metric space]] the (open or closed) balls with centre ''x'' form a neighbourhood base at7 KB (1,205 words) - 09:51, 8 September 2013
- In a [[metric space]] the (open or closed) balls with centre ''x'' form a neighbourhood base at7 KB (1,205 words) - 09:52, 8 September 2013
- {{r|Complete metric space}}462 bytes (60 words) - 16:44, 11 January 2010
- {{r|Metric space}}946 bytes (151 words) - 13:05, 28 December 2008
- {{r|Metric space}}490 bytes (62 words) - 21:05, 11 January 2010
- ...collection of [[open set]]s has a finite subcovering. If the space is a [[metric space]] then compactness is equivalent to the set being [[completeness|complete]]4 KB (652 words) - 14:44, 30 December 2008
- ...anach's fixed-point theorem''' states that a contracting map in a complete metric space has a unique fixed-point. Given a complete metric space (''X'',''ρ''), i.e. a metric space in which every [[Cauchy sequence]] {''x''<sub>''n''</sub>} ⊂ ''X'', i.e.6 KB (996 words) - 06:49, 16 January 2012
- {{r|Complete metric space}}681 bytes (91 words) - 18:06, 11 January 2010
- ...elements '''x''' and '''y''', i.e., a Euclidean space is an example of a [[metric space]].9 KB (1,403 words) - 02:22, 14 October 2013
- {{r|Metric space}}846 bytes (136 words) - 05:03, 2 November 2008
- {{r|Metric space}}882 bytes (141 words) - 07:05, 2 November 2008
- ...he rationals are characterized topologically as the unique [[countable]] [[metric space]] without [[isolated point]]s. ...d]]. The rational numbers do not form a [[completeness (topology)|complete metric space]]; the [[real numbers]] are the completion of <math>\mathbb{Q}</math>.9 KB (1,446 words) - 08:52, 30 May 2009
- 3. Every [[metric space|metric]] <math>d</math> on <math>X</math> gives rise to a topology on <math ...set is a union of elements of <math>\mathcal B</math>. For example, in a metric space the open balls form a basis for the metric topology. A '''sub-basis''' <ma15 KB (2,586 words) - 16:07, 4 January 2013
- * A '''compactum''' if it is a compact [[metric space]].2 KB (331 words) - 07:47, 30 December 2008
- ====[[Metric space|Metric]] and [[Uniform space|uniform]] spaces==== ...omplete space is isometrically embedded into its completion. Every compact metric space is complete; the real line is non-compact but complete; the open interval <28 KB (4,311 words) - 08:36, 14 October 2010
- ...ace''' generalize the notions of a metric (''distance function'') and a [[metric space]] respectively. As a human activity, the theory of uniform spaces is a chap For two points of a metric space, their distance is given, and it is a measure of how close each of the give45 KB (7,747 words) - 06:00, 17 October 2013
- ...lly, the reals are [[completeness (topology)|complete]] (in the sense of [[metric space]]s or [[uniform space]]s, which is a different sense than the Dedekind comp ...ted and better known notion for [[metric space]]s, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not19 KB (2,948 words) - 10:07, 28 February 2024
- ...is a way of defining a possibly fractional exponent for all figures in a [[metric space]] such that the dimension partially describes the amount that the set fills Let ''d'' be a non-negative real number and ''S'' ⊂ ''X'' a subset of a metric space (''X'',''ρ''). The ''d''-dimensional Hausdorff measure of scale ''δ''>015 KB (2,549 words) - 09:18, 17 February 2012
- ...two [[group|group homomorphisms]], two [[vector space|linear maps]], two [[metric space|isometries]], two [[manifold|diffeomorphisms]] etc etc) will preserve the s7 KB (1,151 words) - 14:44, 26 December 2013
- ...orresponding qualitative theory of the eigenfunctions and their [[Complete metric space|completeness]] in a suitable [[function space]] became known as '''Sturm–15 KB (2,332 words) - 04:52, 18 October 2009
- ...his notion has led to the fundamental mathematical concept of a [[complete metric space]]. The [[Cauchy condition]] for the convergence of [[series (mathematics)|s20 KB (3,286 words) - 12:52, 24 August 2013