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In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the uniform norm (or sup norm) assigns to real- or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
-valued
bounded function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s ''f'' defined on a set ''S'' the non-negative number :\, f\, _\infty=\, f\, _=\sup\left\. This
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the
supremum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
is in fact the maximum, the max norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
derived from the uniform norm if and only if f_n converges to f uniformly. The metric generated by this norm is called the Chebyshev metric, after
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian Russian refers to anything related to Russia, including: *Russians (русские, ''r ...
, 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 extended metric still allows one to define a topology on the function space in question. If ''f'' is a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
on a
closed interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, or more generally a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
set, then it is bounded and the
supremum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

supremum
in the above definition is attained by the Weierstrass
extreme value theorem 300px, A continuous function f(x) on the closed interval _showing_the_absolute_max_(red)_and_the_absolute_min_(blue)..html" ;"title=",b/math> showing the absolute max (red) and the absolute min (blue).">,b/math> showing the absolute max (red) an ...
, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, if x is some vector such that x = (x_1, x_2, \ldots ,x_n) in
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
al coordinate space, it takes the form: \, x\, _\infty := \max \left( \left, x_1\ , \ldots , \left, x_n\\right). The set of vectors whose infinity norm is a given constant, , forms the surface of a
hypercube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypercube
with edge length 2''c''. The reason for the subscript "∞" is that whenever ''f'' is continuous :\lim_\, f\, _p=\, f\, _\infty, where :\, f\, _p=\left(\int_D \left, f\^p\,d\mu\right)^ where ''D'' is the domain of ''f'' and the integral amounts to a sum if ''D'' is a
discrete set Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual. Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic c ...
(see
''p''-norm
''p''-norm
). The binary function :d(f,g)=\, f-g\, _\infty is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence converges uniformly to a function ''f'' if and only if :\lim_\, f_n-f\, _\infty=0.\, 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 theoremIn mathematical analysis, the Weierstrass approximation theorem states that every continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
is that the set of all continuous functions on ,b/math> is the uniform closure of the set of polynomials on ,b/math>. For complex
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
functions over a compact space, this turns it into a
C* algebra In mathematics, specifically in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
.


See also

*
Uniform continuity In mathematics, a function (mathematics), function ''f'' is uniformly continuous if, roughly speaking, it is possible to guarantee that ''f''(''x'') and ''f''(''y'') be as close to each other as we please by requiring only that ''x'' and ''y'' be s ...
*
Uniform space In the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathemati ...
*
Chebyshev distance In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...


References

{{DEFAULTSORT:Uniform Norm Functional analysis Norms (mathematics)