In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the uniform norm (or ) assigns to
real- or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
bounded function
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. ...
s defined on a
set the non-negative number
:
This
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
is also called the , the , the , or, when the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions converges to under the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
derived from the uniform norm
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
converges to
uniformly.
If is a
continuous function on a
closed and bounded interval, or more generally a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
set, then it is bounded and the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
in the above definition is attained by the Weierstrass
extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> ...
, so we can replace the supremum by the maximum. In this case, the norm is also called the .
In particular, if is some vector such that
in
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
dimensional
coordinate space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
, it takes the form:
:
Metric and topology
The metric generated by this norm is called the , after
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebysh ...
, 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.
The binary function
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
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitra ...
to a function
if and only if
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
For instance, one restatement of the
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the s ...
is that the set of all continuous functions on
, b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline of ...
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 g ...
functions over a compact space, this turns it into a
C* algebra.
Properties
The set of vectors whose infinity norm is a given constant,
c, forms the surface of a
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...
with edge length
2 c.
The reason for the subscript "
\infty" is that whenever
f is continuous
\lim_\, f\, _p = \, f\, _\infty,
where
\, f\, _p = \left(\int_D , 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
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equi ...
(see
Norm (mathematics)#p-norm, ''p''-norm).
See also
*
*
*
*
References
{{DEFAULTSORT:Uniform Norm
Banach spaces
Functional analysis
Normed spaces
Norms (mathematics)