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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}}
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{{Refimprove|date=December 2009}}
[[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.]]
 
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
 
:<math>\|f\|_\infty=\|f\|_{\infty,S}=\sup\left\{\,\left|f(x)\right|:x\in S\,\right\}.</math>
 
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>
 
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.
 
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'''.
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
 
:<math>\|x\|_\infty=\max\{ |x_1|, \dots, |x_n| \}.</math><!-- avoiding "\," should allow HTML display -->
 
The reason for the subscript "∞" is that whenever ''f'' is continuous
 
:<math>\lim_{p\rightarrow\infty}\|f\|_p=\|f\|_\infty,</math>
 
where
 
:<math>\|f\|_p=\left(\int_D \left|f\right|^p\,d\mu\right)^{1/p}</math>
 
where ''D'' is the domain of ''f'' (and the integral amounts to a sum if ''D'' is a [[discrete set]]).
 
The binary function
 
:<math>d(f,g)=\|f-g\|_\infty</math>
 
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
 
:<math>\lim_{n\rightarrow\infty}\|f_n-f\|_\infty=0.\,</math>
 
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>.
 
For [[complex number|complex]] [[continuity (topology)|continuous]] functions over a [[compact space]], this turns it into a [[C-star algebra|C* algebra]].
 
==See also==
*[[Chebyshev distance]]
*[[Uniform continuity]]
*[[Uniform space]]
 
==References==
 
{{Reflist}}
 
{{DEFAULTSORT:Uniform Norm}}
[[Category:Functional analysis]]
[[Category:Norms (mathematics)]]

Revision as of 00:34, 5 March 2014

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