Borel conjecture - Revision history

66.59.127.126 at 08:04, 26 September 2014

2014-09-26T08:04:43Z

← Older revision		Revision as of 10:04, 26 September 2014
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	~~In [[mathematics]], a [[sequence]]~~ of ~~[[function (mathematics)\|function]]s <math>\{f_{n}\}</math> from a set ''S'' to a metric space ''M''~~ is ~~said to be '''uniformly Cauchy''' if:~~		The name of the author is Nestor. Delaware is the place I love most but I require to transfer for my family. The preferred hobby for my kids and me is dancing and now I'm attempting to make cash with it. Managing people is what I do in my working day occupation.<br><br>my webpage; extended auto warranty; [http://Www.Spa-It.al/ActivityFeed/MyProfile/tabid/60/userId/11049/Default.aspx simply click the following internet page],

	* For all <math>\varepsilon > 0</math>, there exists <math>N>0</math> such that for all <math>x\in S</math>: <math>d(f_{n}(x), f_{m}(x)) < \varepsilon</math> whenever <math>m, n > N</math>.

	~~Another way of saying this~~ is ~~that <math>d_u (f_{n}, f_{m}) \to 0</math> as <math>m, n \to \infty</math>, where~~ the ~~uniform distance <math>d_u</math> between two functions is defined by~~

	~~:<math>d_{u} (f, g) := \sup_{x \in S} d (f(x), g(x)).</math>~~

	~~== Convergence criteria ==~~
	~~A sequence of functions {''f''<sub>n</sub>} from ''S''~~ to ~~''M'' is '''pointwise''' Cauchy if,~~ for ~~each ''x'' ∈ ''S'', the sequence {''f''<sub>n</sub>(''x'')} is a [[Cauchy sequence]] in ''M''~~. ~~This~~ is a weaker condition than being uniformly Cauchy. Nevertheless, if the metric space ''M'' is [[complete metric space\|complete]], then any pointwise Cauchy sequence converges pointwise to a function from ''S'' to ~~''M''~~. ~~Similarly, any uniformly Cauchy sequence will tend [[uniform convergence\|uniformly]] to such a function.~~

	~~The uniform Cauchy property~~ is ~~frequently used when the ''S'' is not just a set, but a [[topological space]], and ''M'' is a complete metric space~~. ~~The following theorem holds:~~

	* Let ''S'' be a topological space and ''M'' a complete metric space. Then any uniformly Cauchy sequence of [[continuous function]]s ''f''<~~sub~~>n<~~/sub~~> ~~: ''S'' &rarr~~; ~~''M'' tends~~ [~~[uniform convergence\|uniformly]] to a unique continuous function ''f''~~ : ~~''S'' → ''M''~~.

	~~== Generalization to uniform spaces ==~~

	~~A [[sequence]] of [[function (mathematics)\|function]]s <math>\{f_{n}\}<~~/~~math> from a set ''S'' to a metric space ''U'' is said to be '''uniformly Cauchy''' if:~~

	* For all <math>x\in S</~~math> and for any [[Uniform space\|entourage]] <math>\varepsilon<~~/~~math>, there exists <math>N>0<~~/~~math> such that <math>d(f_{n}(x), f_{m}(x)) < \varepsilon<~~/~~math> whenever <math>m, n > N<~~/~~math>~~.

	~~==See also==~~
	*[[Modes of convergence (annotated index)]]

	~~[[Category:Functional analysis]]~~
	~~[[Category:Convergence (mathematics)]]~~

	~~{{mathanalysis-stub}}~~

en>Giftlite: −sp

2013-03-18T17:11:09Z

−sp

← Older revision		Revision as of 19:11, 18 March 2013
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	~~Greetings! I am Marvella and I really feel comfortable when individuals use over the counter std test;~~ [~~http~~://~~dore~~.~~gia~~.~~ncnu.edu.tw~~/~~88ipart~~/~~node~~/~~1326254 read this post here~~], ~~full title~~. ~~My day occupation~~ is a ~~meter reader~~. ~~To gather cash is~~ a ~~factor that I~~'~~m totally addicted~~ to. ~~My family life~~ in ~~Minnesota~~ and ~~my family members enjoys it~~.		In [[mathematics]], a [[sequence]] of [[function (mathematics)\|function]]s <math>\{f_{n}\}</math> from a set ''S'' to a metric space ''M'' is said to be '''uniformly Cauchy''' if:

			* For all <math>\varepsilon > 0</math>, there exists <math>N>0</math> such that for all <math>x\in S</math>: <math>d(f_{n}(x), f_{m}(x)) < \varepsilon</math> whenever <math>m, n > N</math>.

			Another way of saying this is that <math>d_u (f_{n}, f_{m}) \to 0</math> as <math>m, n \to \infty</math>, where the uniform distance <math>d_u</math> between two functions is defined by

			:<math>d_{u} (f, g) := \sup_{x \in S} d (f(x), g(x)).</math>

			== Convergence criteria ==
			A sequence of functions {''f''<sub>n</sub>} from ''S'' to ''M'' is '''pointwise''' Cauchy if, for each ''x'' ∈ ''S'', the sequence {''f''<sub>n</sub>(''x'')} is a [[Cauchy sequence]] in ''M''. This is a weaker condition than being uniformly Cauchy. Nevertheless, if the metric space ''M'' is [[complete metric space\|complete]], then any pointwise Cauchy sequence converges pointwise to a function from ''S'' to ''M''. Similarly, any uniformly Cauchy sequence will tend [[uniform convergence\|uniformly]] to such a function.

			The uniform Cauchy property is frequently used when the ''S'' is not just a set, but a [[topological space]], and ''M'' is a complete metric space. The following theorem holds:

			* Let ''S'' be a topological space and ''M'' a complete metric space. Then any uniformly Cauchy sequence of [[continuous function]]s ''f''<sub>n</sub> : ''S'' → ''M'' tends [[uniform convergence\|uniformly]] to a unique continuous function ''f'' : ''S'' → ''M''.

			== Generalization to uniform spaces ==

			A [[sequence]] of [[function (mathematics)\|function]]s <math>\{f_{n}\}</math> from a set ''S'' to a metric space ''U'' is said to be '''uniformly Cauchy''' if:

			* For all <math>x\in S</math> and for any [[Uniform space\|entourage]] <math>\varepsilon</math>, there exists <math>N>0</math> such that <math>d(f_{n}(x), f_{m}(x)) < \varepsilon</math> whenever <math>m, n > N</math>.

			==See also==
			*[[Modes of convergence (annotated index)]]

			[[Category:Functional analysis]]
			[[Category:Convergence (mathematics)]]

			{{mathanalysis-stub}}

en>Aleph4: /* The origin of the conjecture */ there is more than one famous Borel

2011-05-07T18:21:18Z

The origin of the conjecture: there is more than one famous Borel

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Greetings! I am Marvella and I really feel comfortable when individuals use over the counter std test; [http://dore.gia.ncnu.edu.tw/88ipart/node/1326254 read this post here], full title. My day occupation is a meter reader. To gather cash is a factor that I'm totally addicted to. My family life in Minnesota and my family members enjoys it.