Editing Uniform norm (section)

{{Short description|Function in mathematical analysis}}
{{About|the function space norm|the finite-dimensional vector space distance|Chebyshev distance|the uniformity norm in additive combinatorics|Gowers norm}}
{{Refimprove|date=December 2009}}
[[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]], the '''uniform norm''' (or '''{{visible anchor|sup norm}}''') assigns, to [[Real number|real-]] or [[Complex number|complex]]-valued [[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]] {{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]] 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]], 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]], 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]].