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{{Short description|Function in mathematical analysis}}
[[Image:Vector norm sup.svg|frame|right|The perimeter of the square is the set of points in {{math|ℝ{{sup|2}}}} where the sup norm equals a fixed positive constant. For example, points {{math|(2, 0)}}, {{math|(2, 1)}}, and {{math|(2, 2)}} lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2.]]

In [[Mathematical analysis|mathematical analysis]], the '''uniform norm''' (or '''{{visible anchor|sup norm}}''') assigns to [[Real number|real-]] or [[Complex number|complex]]-valued [[Bounded function|bounded function]]s {{tmath|f}} defined on a [[Set (mathematics)|set]] {{tmath|S}} the non-negative number
:<math>\|f\|_\infty = \|f\|_{\infty,S} = \sup\left\{\,|f(s)| : s \in S\,\right\}.</math>

This [[Norm (mathematics)|norm]] is also called the '''{{visible anchor|supremum norm}},''' the '''{{visible anchor|Chebyshev norm}},''' the '''{{visible anchor|infinity norm}},''' or, when the [[Infimum and supremum|supremum]] is in fact the maximum, the '''{{visible anchor|max norm}}'''.  The name "uniform norm" derives from the fact that a sequence of functions {{tmath|\left\{f_n\right\} }} converges to {{tmath|f}} under the [[Metric (mathematics)|metric]] derived from the uniform norm [[If and only if|if and only if]] {{tmath|f_n}} converges to {{tmath|f}} [[Uniform convergence|uniformly]].<ref>{{cite book|last=Rudin|first=Walter|title=Principles of Mathematical Analysis|url=https://archive.org/details/principlesofmath00rudi|url-access=registration|year=1964|publisher=McGraw-Hill|location=New York|isbn=0-07-054235-X|pages=[https://archive.org/details/principlesofmath00rudi/page/151 151]}}</ref>

If {{tmath|f}} is a [[Continuous function|continuous function]] on a [[Interval (mathematics)|closed and bounded 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|extreme value theorem]], so we can replace the supremum by the maximum. In this case, the norm is also called the '''{{visible anchor|maximum norm}}'''.
In particular, if {{tmath|x}} is some vector such that <math>x = \left(x_1, x_2, \ldots, x_n\right) </math> in [[Finite set|finite]] dimensional [[Coordinate space|coordinate space]], it takes the form:
:<math>\|x\|_\infty := \max \left(\left|x_1\right| , \ldots , \left|x_n\right|\right).</math>
This is called the [[L-infinity|<math>\ell^\infty</math>-norm]].

==Metric and topology==

The metric generated by this norm is called the '''{{visible anchor|Chebyshev metric}}''', after [[Biography:Pafnuty Chebyshev|Pafnuty Chebyshev]], who was first to systematically study it.

If we allow unbounded functions, this formula does not yield a norm or 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.

The binary function
<math display=block>d(f, g) = \|f - g\|_\infty</math>
is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence <math>\left\{f_n : n = 1, 2, 3, \ldots\right\}</math> [[Uniform convergence|converges uniformly]] to a function <math>f</math> if and only if
<math display=block>\lim_{n\rightarrow\infty} \left\|f_n - f\right\|_\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 <math>A.</math> 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 continuous functions over a compact space, this turns it into a C* algebra.

==Properties==

The set of vectors whose infinity norm is a given constant, <math>c,</math> forms the surface of a [[Hypercube|hypercube]] with edge length&nbsp;<math>2 c.</math>

The reason for the subscript “<math>\infty</math>” is that whenever <math>f</math> is continuous and <math>\Vert f \Vert_p < \infty</math> for some <math>p \in (0, \infty)</math>, then
<math display=block>\lim_{p \to \infty}\|f\|_p = \|f\|_\infty,</math>
where
<math display=block>\|f\|_p = \left(\int_D |f|^p\,d\mu\right)^{1/p}</math>
where <math>D</math> is the domain of <math>f</math>; the integral amounts to a sum if <math>D</math> is a discrete set (see [[Norm (mathematics)#p-norm|''p''-norm]]).

==See also==

* {{annotated link|L-infinity}}
* {{annotated link|Uniform continuity}}
* {{annotated link|Uniform space}}
* {{annotated link|Chebyshev distance}}

==References==

{{reflist}}

{{Functional Analysis}}

{{DEFAULTSORT:Uniform Norm}}

[[Category:Banach spaces]]
[[Category:Functional analysis]]
[[Category:Normed spaces]]
[[Category:Norms (mathematics)]]

{{Sourceattribution|Uniform norm}}