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| {{About|the function space norm|the finite-dimensional vector space distance|Chebyshev distance|the uniformity norm in additive combinatorics|Gowers norm}}
| | They call me Kelvin but it's not the most masculine person's name. Since she was 18 she's been working as the production and planning officer but soon her husband and her will start their own family based business. Mississippi wherever our house is. Playing handball is one challenge that I did for years. Go to her website to discover more: http://Wmx6.Newsvine.com/_news/2014/06/15/24330782-tolerancia-y-implementaciones-varias-de-la-resina-epoxi<br><br>my homepage: [http://Wmx6.Newsvine.com/_news/2014/06/15/24330782-tolerancia-y-implementaciones-varias-de-la-resina-epoxi resinas] |
| {{Refimprove|date=December 2009}}
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| [[Image:Vector norm sup.svg|frame|right|The black square is the set of points in '''R'''<sup>2</sup> where the sup norm equals a fixed positive constant.]]
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| In [[mathematical analysis]], the '''uniform [[norm (mathematics)|norm]]''' (or '''sup norm''') assigns to [[real number|real-]] or [[complex number|complex]]-valued bounded functions ''f'' defined on a set ''S'' the non-negative number
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| :<math>\|f\|_\infty=\|f\|_{\infty,S}=\sup\left\{\,\left|f(x)\right|:x\in S\,\right\}.</math> | |
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| This norm is also called the '''[[supremum]] norm,''' the '''Chebyshev norm,''' or the '''infinity norm.''' The name "uniform norm" derives from the fact that a sequence of functions <math>\{f_n\}</math> converges to f under the metric derived from the uniform norm if and only if <math>f_n</math> converges to <math>f</math> [[uniform convergence|uniformly]].<ref>{{cite book|last=Rudin|first=Walter|title=Principles of Mathematical Analysis|year=1964|publisher=McGraw-Hill|location=New York|isbn=0-07-054235-X|pages=151}}</ref>
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| If we allow unbounded functions, this formula does not yield a norm or [[metric (mathematics)|metric]] in a strict sense, although the obtained so-called [[metric (mathematics)#Generalized metrics|extended metric]] still allows one to define a topology on the function space in question.
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| If ''f'' is a [[continuous function]] on a [[closed interval]], or more generally a [[compact space|compact]] set, then it is bounded and the [[supremum]] in the above definition is attained by the Weierstrass [[extreme value theorem]], so we can replace the supremum by the maximum. In this case, the norm is also called the '''maximum norm'''.
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| In particular, for the case of a vector <math>x=(x_1,\dots,x_n)</math> in [[finite set|finite]] [[dimension]]al [[coordinate space]], it takes the form
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| :<math>\|x\|_\infty=\max\{ |x_1|, \dots, |x_n| \}.</math><!-- avoiding "\," should allow HTML display -->
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| The reason for the subscript "∞" is that whenever ''f'' is continuous
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| :<math>\lim_{p\rightarrow\infty}\|f\|_p=\|f\|_\infty,</math>
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| where
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| :<math>\|f\|_p=\left(\int_D \left|f\right|^p\,d\mu\right)^{1/p}</math> | |
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| where ''D'' is the domain of ''f'' (and the integral amounts to a sum if ''D'' is a [[discrete set]]).
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| The binary function
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| :<math>d(f,g)=\|f-g\|_\infty</math> | |
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| is then a [[metric (mathematics)|metric]] on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence { ''f''<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } [[uniform convergence|converges uniformly]] to a function ''f'' if and only if
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| :<math>\lim_{n\rightarrow\infty}\|f_n-f\|_\infty=0.\,</math>
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| We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called ''uniformly closed'' and closures ''uniform closures''. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on A. For instance, one restatement of the [[Stone–Weierstrass theorem]] is that the set of all continuous functions on <math>[a,b]</math> is the uniform closure of the set of polynomials on <math>[a,b]</math>.
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| For [[complex number|complex]] [[continuity (topology)|continuous]] functions over a [[compact space]], this turns it into a [[C-star algebra|C* algebra]].
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| ==See also==
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| *[[Chebyshev distance]]
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| *[[Uniform continuity]]
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| *[[Uniform space]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Uniform Norm}}
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| [[Category:Functional analysis]]
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| [[Category:Norms (mathematics)]]
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They call me Kelvin but it's not the most masculine person's name. Since she was 18 she's been working as the production and planning officer but soon her husband and her will start their own family based business. Mississippi wherever our house is. Playing handball is one challenge that I did for years. Go to her website to discover more: http://Wmx6.Newsvine.com/_news/2014/06/15/24330782-tolerancia-y-implementaciones-varias-de-la-resina-epoxi
my homepage: resinas