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 ( ...
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 cal ...
of a given family of
real-valued
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s defined on a
closed and
bounded interval has a
uniformly convergent 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 the theory of
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
,
Montel's theorem in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, and the
Peter–Weyl theorem in
harmonic analysis 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 and
Giulio Ascoli. 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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. 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 Hausdorff space into a
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
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 ...
; see .
Statement and first consequences
By definition, a sequence
of
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
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 cal ...
of real-valued continuous functions defined on a closed and bounded
interval of the
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
. If this sequence is
uniformly bounded and uniformly
equicontinuous, 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.
: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 functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the
mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
that for all and ,
:
where is the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
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 functions. Suppose that the functions are continuously differentiable with derivatives . Suppose that 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
n'' is
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...
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 on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that
, f(x) - f(y) , \leq C\, x - y\, ^
for all and in the do ...
of order , , with a fixed constant ,
::