View source for Uniform norm

{{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]].

== Definition ==
Uniform norms are defined, in general, for [[bounded function]]s valued in a [[normed space]]. Let <math>X</math> be a set and let <math>(Y,\|\|_Y)</math> be a [[normed space]]. On the set <math>Y^X</math> of functions from <math>X</math> to <math>Y</math>, there is an [[extended norm]] defined by
:<math>\|f\|=\sup_{x\in X}\|f(x)\|_Y\in[0,\infty].</math>
This is in general an extended norm since the function <math>f</math> may not be bounded. Restricting this extended norm to the bounded functions (i.e., the functions with finite above extended norm) yields a (finite-valued) norm, called the '''uniform norm''' on <math>Y^X</math>. Note that the definition of uniform norm does not rely on any additional structure on the set <math>X</math>, although in practice <math>X</math> is often at least a [[topological space]].

The convergence on <math>Y^X</math> in the topology induced by the uniform extended norm is the [[uniform convergence]], for sequences, and also for [[net (mathematics)|nets]] and [[filter (mathematics)|filters]] on <math>Y^X</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 function (topology)|continuous]] functions over a compact space, this turns it into a [[C-star algebra|C* algebra]] (cf. [[Gelfand representation]]).

== Weaker structures inducing the topology of uniform convergence ==
=== Uniform metric ===
{{Main|Chebyshev distance}}
The '''uniform metric''' between two bounded functions <math>f,g\colon X\to Y</math> from a set <math>X</math> to a [[metric space]] <math>(Y,d_Y)</math> is defined by
:<math>d(f,g)=\sup_{x\in X}d_Y(f(x),g(x))</math>
The uniform metric is also called the '''{{visible anchor|Chebyshev metric}}''', after [[Pafnuty Chebyshev]], who was first to systematically study it. In this case, <math>f</math> is bounded precisely if <math>d(f,g)</math> is finite for some [[constant function]] <math>g</math>. 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 convergence is then still the [[uniform convergence]]. In particular, 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}d(f_n,f)= 0.\,</math>

If <math>(Y,\|\|_Y)</math> is a [[normed space]], then it is a [[metric space]] in a natural way. The extended metric on <math>Y^X</math> induced by the uniform extended norm is the same as the uniform extended metric
:<math>d(f,g)=\sup_{x\in X}\|f(x)-g(x)\|_Y</math>
on <math>Y^X</math>

=== Uniformity of uniform convergence ===
{{See also|Topologies on spaces of linear maps}}
Let <math>X</math> be a set and let <math>(Y,\mathcal E_Y)</math> be a [[uniform space]]. A sequence <math>(f_n)</math> of functions from <math>X</math> to <math>Y</math> is said to converge uniformly to a function <math>f</math> if for each entourage <math>E\in\mathcal E_Y</math> there is a natural number <math>n_0</math> such that, <math>(f_n(x),f(x))</math> belongs to <math>E</math> whenever <math>x\in X</math> and <math>n\ge n_0</math>. Similarly for a net. This is a convergence in a topology on <math>Y^X</math>. In fact, the sets
:<math>\{(f,g)\colon\forall x\in X\colon(f(x),g(x))\in E\}</math>
where <math>E</math> runs through entourages of <math>Y</math> form a fundamental system of entourages of a uniformity on <math>Y^X</math>, called the '''uniformity of uniform convergence''' on <math>Y^X</math>. The uniform convergence is precisely the convergence under its uniform topology.

If <math>(Y,d_Y)</math> is a [[metric space]], then it is by default equipped with the [[metric uniformity]]. The metric uniformity on <math>Y^X</math> with respect to the uniform extended metric is then the uniformity of uniform convergence on <math>Y^X</math>.

==Properties==

The set of vectors whose infinity norm is a given constant, <math>c,</math> forms the surface of a [[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}}

{{Lp spaces}}
{{Banach spaces}}
{{Functional Analysis}}

{{DEFAULTSORT:Uniform Norm}}

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