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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 :\left, f_n(x)\ \le M 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 :\left, f_n(x)-f_n(y)\ < \varepsilon 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 , :\left, f_n(x) - f_n(y)\ \le K , x-y, , 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 ''fn'' 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 : ::\left, f_n(x) - f_n(y)\ \le K, x-y, :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 , ::\left, f(x) - f(y)\ \le M \, , x - y, ^\alpha, \qquad x, y \in , b/math> :is relatively compact in . In particular, the unit ball of the Hölder space is compact in . This holds more generally for scalar functions on a compact metric space satisfying a Hölder condition with respect to the metric on .


Generalizations


Euclidean spaces

The Arzelà–Ascoli theorem holds, more generally, if the functions take values in -dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, and the proof is very simple: just apply the -valued version of the Arzelà–Ascoli theorem times to extract a subsequence that converges uniformly in the first coordinate, then a sub-subsequence that converges uniformly in the first two coordinates, and so on. The above examples generalize easily to the case of functions with values in Euclidean space.


Compact metric spaces and compact Hausdorff spaces

The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary compact
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 ...
s and, more generally still,
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 ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
s. Let ''X'' be a compact Hausdorff space, and let ''C''(''X'') be the space 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 on ''X''. A subset is said to be ''equicontinuous'' if for every ''x'' ∈ ''X'' and every , ''x'' has a neighborhood ''Ux'' such that :\forall y \in U_x, \forall f \in \mathbf : \qquad , f(y) - f(x), < \varepsilon. A set is said to be ''pointwise bounded'' if for every ''x'' ∈ ''X'', :\sup \ < \infty. A version of the Theorem holds also in the space ''C''(''X'') of real-valued continuous functions on 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 ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
''X'' : :Let ''X'' be a compact Hausdorff space. Then a subset F of ''C''(''X'') is relatively compact in the topology induced by the uniform norm
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is equicontinuous and pointwise bounded. The Arzelà–Ascoli theorem is thus a fundamental result in the study of the algebra of continuous functions on a compact Hausdorff space. Various generalizations of the above quoted result are possible. For instance, the functions can assume values in a metric space or (Hausdorff)
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
with only minimal changes to the statement (see, for instance, , ): :Let ''X'' be a compact Hausdorff space and ''Y'' a metric space. Then is 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 ...
if and only if it is equicontinuous, pointwise relatively compact and closed. Here pointwise relatively compact means that for each ''x'' ∈ ''X'', the set is relatively compact in ''Y''. In the case that ''Y'' is complete, the proof given above can be generalized in a way that does not rely on the separability of the domain. On a compact Hausdorff space ''X'', for instance, the equicontinuity is used to extract, for each ε = 1/''n'', a finite open covering of ''X'' such that the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of any function in the family is less than ε on each
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
in the cover. The role of the rationals can then be played by a set of points drawn from each open set in each of the countably many covers obtained in this way, and the main part of the proof proceeds exactly as above. A similar argument is used as a part of the proof for the general version which does not assume completeness of ''Y''.


Functions on non-compact spaces

The Arzela-Ascoli theorem generalises to functions X \rightarrow Y where X is not compact. Particularly important are cases where X is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. Recall that if X is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
and Y is 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 ...
(such as any metric space or any
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
, metrisable or not), there is the topology of compact convergence on the set \mathfrak(X,Y) of functions X \rightarrow Y; it is set up so that a sequence (or more generally a filter or net) of functions converges if and only if it converges ''uniformly'' on each compact subset of X. Let \mathcal_c(X,Y) be the subspace of \mathfrak(X,Y) consisting of continuous functions, equipped with the topology of compact convergence. Then one form of the Arzelà-Ascoli theorem is the following: :Let X be a topological space, Y a Hausdorff uniform space and H\subset\mathcal_c(X,Y) an equicontinuous set of continuous functions such that \ is relatively compact in Y for each x\in X. Then H is relatively compact in \mathcal_c(X,Y). This theorem immediately gives the more specialised statements above in cases where X is compact and the uniform structure of Y is given by a metric. There are a few other variants in terms of the topology of precompact convergence or other related topologies on \mathfrak(X,Y). It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of X by compact subsets. For details one can consult Bourbaki (1998), Chapter X, § 2, nr 5.


Non-continuous functions

Solutions of numerical schemes for parabolic equations are usually piecewise constant, and therefore not continuous, in time. As their jumps nevertheless tend to become small as the time step goes to 0, it is possible to establish uniform-in-time convergence properties using a generalisation to non-continuous functions of the classical Arzelà–Ascoli theorem (see e.g. ). Denote by S(X,Y) the space of functions from X to Y endowed with the uniform metric :d_S(v,w)=\sup_d_Y(v(t),w(t)). Then we have the following: :Let X be a compact metric space and Y a complete metric space. Let \_ be a sequence in S(X,Y) such that there exists a function \omega:X\times X\to ,\infty/math> and a sequence \_\subset oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
over an open subset ''U'' ⊂ ''X'' is less than ''ε'': :N(\varepsilon, U) = \. 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.


Further examples

* To every function that is
-integrable on , with , associate the function defined on by ::G(x) = \int_0^x g(t) \, \mathrmt. :Let be the set of functions corresponding to functions in the unit ball of the space . If is the Hölder conjugate of , defined by , then
Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces. The numbers an ...
implies that all functions in satisfy a Hölder condition with and constant . :It follows that is compact in . This means that the correspondence defines 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 ...
Linear map">linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
between the Banach spaces and . Composing with the injection of into , one sees that acts compactly from to itself. The case can be seen as a simple instance of the fact that the injection from the Sobolev space H^1_0(\Omega) into , for a bounded open set in , is compact. *When is a compact linear operator from a Banach space to a Banach space , its
transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
is compact from the (continuous) dual to . This can be checked by the Arzelà–Ascoli theorem. :Indeed, the image of the closed unit ball of is contained in a compact subset of . The unit ball of defines, by restricting from to , a set of (linear) continuous functions on that is bounded and equicontinuous. By Arzelà–Ascoli, for every sequence in , there is a subsequence that converges uniformly on , and this implies that the image T^*(y^*_) of that subsequence is Cauchy in . *When is holomorphic in an open disk , with modulus bounded by , then (for example by Cauchy's formula) its derivative has modulus bounded by in the smaller disk If a family of holomorphic functions on is bounded by on , it follows that the family of restrictions to is equicontinuous on . Therefore, a sequence converging uniformly on can be extracted. This is a first step in the direction of Montel's theorem. * Let C( ,TL^1(\mathbb^N)) be endowed with the uniform metric \textstyle\sup_\, v(\cdot,t)-w(\cdot,t)\, _. Assume that u_n=u_n(x,t)\subset C( ,TL^1(\mathbb^N)) is a sequence of solutions of a certain
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
(PDE), where the PDE ensures the following a priori estimates: x\mapsto u_n(x,t) is equicontinuous for all t, x\mapsto u_n(x,t) is equitight for all t, and, for all (t,t')\in ,Ttimes ,T/math> and all n\in\mathbb, \, u_n(\cdot,t)-u_n(\cdot,t')\, _ is small enough when , t-t', is small enough. Then by the Fréchet–Kolmogorov theorem, we can conclude that \ is relatively compact in L^1(\mathbb^N). Hence, we can, by (a generalization of) the Arzelà–Ascoli theorem, conclude that \ is relatively compact in C( ,TL^1(\mathbb^N)).


See also

* Helly's selection theorem * Fréchet–Kolmogorov theorem


References

* . * . * . * . * * . * . * .
''Arzelà-Ascoli theorem'' at Encyclopaedia of Mathematics
* * * {{DEFAULTSORT:Arzela-Ascoli theorem Articles containing proofs Compactness theorems Theory of continuous functions Theorems in real analysis Theorems in functional analysis Topology of function spaces