en>Eurodyne: Reverted 1 edit by 101.56.114.169 (talk). (TW)

2014-12-21T17:31:24Z

Reverted 1 edit by 101.56.114.169 (talk). (TW)

74.14.18.134 at 02:56, 6 November 2013

2013-11-06T02:56:43Z

← Older revision		Revision as of 04:56, 6 November 2013
Line 1:		Line 1:
	~~Hello~~, ~~my name~~ is ~~Andrew~~ and ~~my spouse doesn~~'t ~~like~~ it at all. ~~He works~~ as a ~~bookkeeper~~. ~~What me~~ and ~~my family members adore~~ is ~~bungee leaping but I~~'~~ve been taking~~ on ~~new issues recently~~. ~~Some time ago he selected~~ to ~~reside in North Carolina and he doesn~~'~~t strategy on altering it~~.<br><br>~~Feel free to surf to my web site~~ :: [~~http~~://~~jplusfn~~.~~gaplus~~.~~kr/xe/qna/78647 clairvoyance~~]		In [[mathematics]], a '''regulated function''' (or '''ruled function''') is a "well-behaved" [[function (mathematics)\|function]] of a single [[real number\|real]] variable. Regulated functions arise as a class of [[integration (mathematics)\|integrable functions]], and have several equivalent characterisations. Regulated functions were introduced by Georg Aumann in 1954; the corresponding [[regulated integral]] was promoted by the [[Bourbaki]] group, including [[Jean Dieudonné]].

			==Definition==

			Let ''X'' be a [[Banach space]] with norm \|\| - \|\|<sub>''X''</sub>. A function ''f'' : [0, ''T''] → ''X'' is said to be a '''regulated function''' if one (and hence both) of the following two equivalent conditions holds true {{harv\|Dieudonné\|1969\|loc=§7.6}}:

			* for every ''t'' in the [[interval (mathematics)\|interval]] [0, ''T''], both the [[limit of a function\|left and right limits]] ''f''(''t''−) and ''f''(''t''+) exist in ''X'' (apart from, obviously, ''f''(0−) and ''f''(''T''+));

			* there exists a [[sequence (mathematics)\|sequence]] of [[step function]]s ''φ''<sub>''n''</sub> : [0, ''T''] → ''X'' [[uniform convergence\|converging uniformly]] to ''f'' (i.e. with respect to the [[supremum norm]] \|\| - \|\|<sub>∞</sub>).

			It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:

			* for every ''δ'' > 0, there is some step function ''φ''<sub>''δ''</sub> : [0, ''T''] → ''X'' such that

			::<math>\\| f - \varphi_{\delta} \\|_{\infty} = \sup_{t \in [0, T]} \\| f(t) - \varphi_{\delta} (t) \\|_{X} < \delta;</math>

			* ''f'' lies in the [[closed set\|closure]] of the space Step([0, ''T'']; ''X'') of all step functions from [0, ''T''] into ''X'' (taking closure with respect to the supremum norm in the space B([0, ''T'']; ''X'') of all bounded functions from [0, ''T''] into ''X'').

			==Properties of regulated functions==

			Let Reg([0, ''T'']; ''X'') denote the [[Set (mathematics)\|set]] of all regulated functions ''f'' : [0, ''T''] → ''X''.

			* Sums and scalar multiples of regulated functions are again regulated functions. In other words, Reg([0, ''T'']; ''X'') is a [[vector space]] over the same [[field (mathematics)\|field]] '''K''' as the space ''X''; typically, '''K''' will be the [[real number\|real]] or [[complex number]]s. If ''X'' is equipped with an operation of multiplication, then products of regulated functions are again regulated functions. In other words, if ''X'' is a '''K'''-[[Algebra over a field\|algebra]], then so is Reg([0, ''T'']; ''X'').

			* The supremum norm is a [[norm (mathematics)\|norm]] on Reg([0, ''T'']; ''X''), and Reg([0, ''T'']; ''X'') is a [[topological vector space]] with respect to the topology induced by the supremum norm.

			* As noted above, Reg([0, ''T'']; ''X'') is the closure in B([0, ''T'']; ''X'') of Step([0, ''T'']; ''X'') with respect to the supremum norm.

			* If ''X'' is a [[Banach space]], then Reg([0, ''T'']; ''X'') is also a Banach space with respect to the supremum norm.

			* Reg([0, ''T'']; '''R''') forms an infinite-dimensional real [[Banach algebra]]: finite linear combinations and products of regulated functions are again regulated functions.

			* Since a [[continuous function]] defined on a [[compact space]] (such as [0, ''T'']) is automatically [[uniformly continuous function\|uniformly continuous]], every continuous function ''f'' : [0, ''T''] → ''X'' is also regulated. In fact, with respect to the supremum norm, the space ''C''<sup>0</sup>([0, ''T'']; ''X'') of continuous functions is a [[closed set\|closed]] [[linear subspace]] of Reg([0, ''T'']; ''X'').

			* If ''X'' is a [[Banach space]], then the space BV([0, ''T'']; ''X'') of functions of [[bounded variation]] forms a [[dense set\|dense]] linear subspace of Reg([0, ''T'']; ''X''):

			::<math>\mathrm{Reg}([0, T]; X) = \overline{\mathrm{BV} ([0, T]; X)} \mbox{ w.r.t. } \\| \cdot \\|_{\infty}.</math>

			* If ''X'' is a Banach space, then a function ''f'' : [0, ''T''] → ''X'' is regulated [[if and only if]] it is of [[Bounded variation#Weighted BV functions\|bounded ''φ''-variation]] for some ''φ'':

			::<math>\mathrm{Reg}([0, T]; X) = \bigcup_{\varphi} \mathrm{BV}_{\varphi} ([0, T]; X).</math>

			* If ''X'' is a [[separable space\|separable]] [[Hilbert space]], then Reg([0, ''T'']; ''X'') satisfies a compactness theorem known as the [[Fraňková-Helly selection theorem]].

			* The set of [[Discontinuity (mathematics)\|discontinuities]] of a regulated function is [[countable]]: to see this it is sufficient to note that given <math> \epsilon > 0 </math>, the set of points at which the right and left limits differ by more than <math> \epsilon</math> is finite. In particular, the discontinuity set has [[measure zero]], from which it follows that a regulated function has a well-defined [[Riemann integral]].

			* The integral, as defined on step functions in the obvious way, extends naturally to Reg([0, ''T'']; ''X'') by defining the integral of a regulated function to be the limit of the integrals of any sequence of step functions converging uniformly to it. This extension is [[well-defined]] and satisfies all of the usual properties of an integral. In particular, the [[regulated integral]]
			** is a [[bounded linear function]] from Reg([0, ''T'']; ''X'') to ''X''; hence, in the case ''X'' = '''R''', the integral is an element of the [[continuous dual space\|space that is dual]] to Reg([0, ''T'']; '''R''');
			** agrees with the [[Riemann integral]].

			==References==

			* {{citation
			\| first = Georg
			\| last = Aumann
			\| title = Reelle Funktionen
			\| language = German
			\| series = Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd LXVIII
			\| publisher = Springer-Verlag
			\| location = Berlin
			\| year = 1954
			\| pages = viii+416
			}} {{MathSciNet\| id = 0061652}}
			* {{citation
			\| first = Jean
			\| last = Dieudonné
			\| authorlink = Jean Dieudonné
			\| title = Foundations of Modern Analysis
			\| publisher = Academic Press
			\| year = 1969
			\| pages = xviii+387
			}} {{MathSciNet \| id = 0349288}}
			* {{citation
			\| last = Fraňková
			\| first = Dana
			\| title = Regulated functions
			\| journal = Math. Bohem.
			\| volume = 116
			\| year = 1991
			\| pages = 20–59
			\| issn = 0862-7959
			\| issue = 1
			}} {{MathSciNet \| id = 1100424}}
			* {{citation
			\| last = Gordon
			\| first = Russell A.
			\| title = The Integrals of Lebesgue, Denjoy, Perron, and Henstock
			\| series = Graduate Studies in Mathematics, 4
			\| publisher = American Mathematical Society
			\| location = Providence, RI
			\| year = 1994
			\| pages = xii+395
			\| isbn = 0-8218-3805-9
			}} {{MathSciNet \| id = 1288751}}
			* {{citation
			\| last = Lang
			\| first = Serge
			\| authorlink = Serge Lang
			\| title = Differential Manifolds
			\| edition = Second
			\| publisher = Springer-Verlag
			\| location = New York
			\| year = 1985
			\| pages = ix+230
			\| isbn = 0-387-96113-5
			}} {{MathSciNet \| id = 772023}}

			[[Category:Real analysis]]
			[[Category:Types of functions]]

en>Pinethicket: Reverted edits by 128.210.49.204 (talk) to last version by 85.216.208.55

2012-06-27T19:15:56Z

Reverted edits by 128.210.49.204 (talk) to last version by 85.216.208.55

New page

Hello, my name is Andrew and my spouse doesn't like it at all. He works as a bookkeeper. What me and my family members adore is bungee leaping but I've been taking on new issues recently. Some time ago he selected to reside in North Carolina and he doesn't strategy on altering it.<br><br>Feel free to surf to my web site :: [http://jplusfn.gaplus.kr/xe/qna/78647 clairvoyance]

Bicarbonate buffering system - Revision history

en>Eurodyne: Reverted 1 edit by 101.56.114.169 (talk). (TW)

74.14.18.134 at 02:56, 6 November 2013

en>Pinethicket: Reverted edits by 128.210.49.204 (talk) to last version by 85.216.208.55