The Arzelà–Ascoli theorem is a fundamental result of
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 ...
giving
necessary and sufficient conditions
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
to decide whether every
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of a given family of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s defined on a
closed and
bounded interval has a
uniformly convergent
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 arbitrarily s ...
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
. The main condition is the
equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the
Peano existence theorem
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees t ...
in the theory of
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
,
Montel's theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic ...
in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, and the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
and various results concerning compactness of integral operators.
The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians
Cesare Arzelà
Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continu ...
and
Giulio Ascoli
Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was a Jewish-Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868.
In 1872 he became Professor of Algebra and Calculus of the Politec ...
. A weak form of the theorem was proven by , who established the sufficient condition for compactness, and by , who established the necessary condition and gave the first clear presentation of the result. A further generalization of the theorem was proven by , to sets of real-valued continuous functions with domain 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 ...
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. Modern formulations of the theorem allow for the domain to be compact
Hausdorff and for the range to be an arbitrary metric space. More general formulations of the theorem exist that give necessary and sufficient conditions for a family of functions from a
compactly generated In mathematics, compactly generated can refer to:
* Compactly generated group, a topological group which is algebraically generated by one of its compact subsets
*Compactly generated space
In topology, a compactly generated space is a topological s ...
Hausdorff space into a
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
to be compact in the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and ...
; see .
Statement and first consequences
By definition, a sequence
of
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s on an interval is ''uniformly bounded'' if there is a number such that
:
for every function belonging to the sequence, and every . (Here, must be independent of and .)
The sequence is said to be ''uniformly equicontinuous'' if, for every , there exists a such that
:
whenever for all functions in the sequence. (Here, may depend on , but not , or .)
One version of the theorem can be stated as follows:
:Consider a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of real-valued continuous functions defined on a closed and bounded
interval of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
. If this sequence is
uniformly bounded
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.
...
and uniformly
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
, then there exists a
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
that
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 arbitrarily s ...
.
:The converse is also true, in the sense that if every subsequence of itself has a uniformly convergent subsequence, then is uniformly bounded and equicontinuous.
Immediate examples
Differentiable functions
The hypotheses of the theorem are satisfied by a uniformly bounded sequence of
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the
mean value theorem that for all and ,
:
where ''K'' is 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 ...
of the derivatives of functions in the sequence and is independent of . So, given , let to verify the definition of equicontinuity of the sequence. This proves the following corollary:
* Let be a uniformly bounded sequence of real-valued differentiable functions on such that the derivatives are uniformly bounded. Then there exists a subsequence that converges uniformly on .
If, in addition, the sequence of second derivatives is also uniformly bounded, then the derivatives also converge uniformly (up to a subsequence), and so on. Another generalization holds for
continuously differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
s. Suppose that the functions are continuously differentiable with derivatives . Suppose that ''f
n''′ are uniformly equicontinuous and uniformly bounded, and that the sequence is pointwise bounded (or just bounded at a single point). Then there is a subsequence of the converging uniformly to a continuously differentiable function.
The diagonalization argument can also be used to show that a family of infinitely differentiable functions, whose derivatives of each order are uniformly bounded, has a uniformly convergent subsequence, all of whose derivatives are also uniformly convergent. This is particularly important in the theory of distributions.
Lipschitz and Hölder continuous functions
The argument given above proves slightly more, specifically
* If is a uniformly bounded sequence of real valued functions on such that each ''f'' is
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
with the same Lipschitz constant :
::
:for all and all , then there is a subsequence that converges uniformly on .
The limit function is also Lipschitz continuous with the same value for the Lipschitz constant. A slight refinement is
* A set of functions on that is uniformly bounded and satisfies a
Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that
: , f(x) - f(y) , \leq C\, ...
of order , , with a fixed constant ,
::
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. Then
.
Whereas most formulations of the Arzelà–Ascoli theorem assert sufficient conditions for a family of functions to be (relatively) compact in some topology, these conditions are typically also necessary. For instance, if a set F is compact in ''C''(''X''), the Banach space of real-valued continuous functions on a compact Hausdorff space with respect to its uniform norm, then it is bounded in the uniform norm on ''C''(''X'') and in particular is pointwise bounded. Let ''N''(''ε'', ''U'') be the set of all functions in F whose
For a fixed ''x''∈''X'' and ''ε'', the sets ''N''(''ε'', ''U'') form an open covering of F as ''U'' varies over all open neighborhoods of ''x''. Choosing a finite subcover then gives equicontinuity.
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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 ...