Arzelà-Ascoli theorem
Encyclopedia
In mathematics
, the Arzelà–Ascoli theorem of functional analysis
gives necessary and sufficient conditions to decide whether every subsequence of a given sequence of real
-valued continuous function
s defined on a closed
and bounded
interval
has a uniformly convergent subsequence
. The main condition is the equicontinuity
of the sequence of functions. The theorem is a fundamental result in mathematics. In particular, it forms the basis for the proof of the Peano existence theorem
in the theory of ordinary differential equations and Montel's theorem
in complex analysis
. It also plays a decisive role in the proof of the Peter-Weyl theorem.
The notion of equicontinuity was introduced at around the same time by and . 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 metric space . 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
to be compact in the compact-open topology
. .
s on an interval I = [a,b] is uniformly bounded if there is a number M such that
for every function ƒn belonging to the sequence, and every x ∈ [a,b]. The sequence is equicontinuous if, for every ε > 0, there exists a δ > 0 such that
whenever 
for every ƒn belonging to the sequence. Succinctly, a sequence is equicontinuous if and only if all of its elements have the same modulus of continuity
. In simplest terms, the theorem can be stated as follows:
The hypotheses of the theorem are satisfied by a uniformly bounded sequence {ƒn} of differentiable
functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the mean value theorem
that for all x and y,
where K is the supremum
of the derivatives of functions in the sequence and is independent of n. So, given ε > 0, let δ = ε/2K to verify the definition of equicontinuity of the sequence. This proves the following corollary:
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 ƒn are continuously differentiable with derivatives ƒn′. Suppose that ƒn′ are uniformly equicontinuous and uniformly bounded, and that the sequence ƒn is pointwise bounded (or just bounded at a single point). Then there is subsequence of the ƒn converging uniformly to a continuously differentiable function.
Lipschitz and Hölder continuous functions
The argument given above proves slightly more, specifically
The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is
This holds more generally for scalar functions on a compact metric space X satisfying a Hölder condition with respect to the metric on X.
Euclidean spaces
The Arzelà–Ascoli theorem holds, more generally, if the functions ƒn take values in d-dimensional Euclidean space
Rd, and the proof is very simple: just apply the R-valued version of the Arzelà–Ascoli theorem d 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.
Fix an enumeration {xi}i=1,2,3,... of rational numbers in I. Since F is uniformly bounded, the set of points {ƒ(x1)}ƒ∈F is bounded, and hence by the Bolzano-Weierstrass theorem, there is a sequence {ƒn1} of distinct functions in F such that {ƒn1(x1)} converges. Repeating the same argument for the sequence of points {ƒn1(x2)}, there is a subsequence {ƒn2} of {ƒn1} such that {ƒn2(x2)} converges.
By mathematical induction
this process can be continued, and so there is a chain of subsequences
such that, for each k = 1, 2, 3, …, the subsequence {ƒnk} converges at x1,...,xk. Now form the diagonal subsequence {ƒ} whose mth term ƒm is the mth term in the mth subsequence {ƒnm}. By construction, ƒm converges at every rational point of I.
Therefore, given any ε > 0 and rational xk in I, there is an integer such that
Since the family F is equicontinuous, for this fixed ε and for every x in I, there is an open interval Ux containing x such that
for all ƒ ∈ F and all s, t in I such that s, t ∈ Ux.
The collection of intervals Ux, x ∈ I, forms an open cover of I. Since I is compact, this covering admits a finite subcover U1, ..., UJ. There exists an integer K such that each open interval Uj, 1 ≤ j ≤ J, contains a rational xk with 1 ≤ k ≤ K. Finally, for any t ∈ I, there are j and k so that t and xk belong to the same interval Uj. For this choice of k,
for all n, m > N = max{N(ε,x1), ..., N(ε,xK)}. Consequently, the sequence {ƒn} is uniformly Cauchy, and therefore converges to a continuous function, as claimed. This completes the proof.
s and, more generally still, compact Hausdorff space
s. Let X be a compact Hausdorff space, and let C(X) be the space of real-valued continuous function
s on X. A subset is said to be equicontinuous if for every x ∈ X and every ε > 0, x has a neighborhood Ux such that
for all y ∈ Ux and ƒ ∈ F.
A set is said to be pointwise bounded if for every x ∈ X,
A version of this holds also in the space C(X) of real-valued continuous functions on a compact Hausdorff space
X :
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
with only minimal changes to the statement (see, for instance, ):
Here pointwise relatively compact means that for each x ∈ X, the set } is relatively compact in Y.
The proof given 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 of any function in the family is less than ε on each open set 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.
form an open covering of F, as U varies over open neighborhoods of x. Choosing a finite subcover then gives equicontinuity.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Arzelà–Ascoli theorem of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
gives necessary and sufficient conditions to decide whether every subsequence of a given sequence of real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s defined on a closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
has a uniformly convergent subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
. The main condition is the equicontinuity
Equicontinuity
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...
of the sequence of functions. The theorem is a fundamental result in mathematics. In particular, it forms the basis for the proof of the Peano existence theorem
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 the existence of solutions to certain initial value...
in the theory of ordinary differential equations and Montel's theorem
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 Paul Montel, and give conditions under which a family of holomorphic functions is normal....
in complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
. It also plays a decisive role in the proof of the Peter-Weyl theorem.
The notion of equicontinuity was introduced at around the same time by and . 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 metric space . 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
Compactly generated space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:Equivalently, one can replace closed with open in this definition...
Hausdorff space into a uniform space
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 which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
to be compact in the compact-open topology
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 functional analysis...
. .
Statement and first consequences
A sequence {ƒn}n∈N of continuous functionContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s on an interval I = [a,b] is uniformly bounded if there is a number M such that
for every function ƒn belonging to the sequence, and every x ∈ [a,b]. The sequence is equicontinuous if, for every ε > 0, there exists a δ > 0 such that
whenever for every ƒn belonging to the sequence. Succinctly, a sequence is equicontinuous if and only if all of its elements have the same modulus of continuity
Modulus of continuity
In mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...
. In simplest terms, the theorem can be stated as follows:
- Consider a sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of real-valued continuous functions (ƒn)n∈N defined on a closed and bounded intervalInterval (mathematics)In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[a, b] of the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequenceSubsequenceIn mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
(ƒnk) that converges uniformly.
Examples
Differentiable functionsThe hypotheses of the theorem are satisfied by a uniformly bounded sequence {ƒn} of differentiable
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the mean value theorem
Mean value theorem
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
that for all x and y,
where K is the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of the derivatives of functions in the sequence and is independent of n. So, given ε > 0, let δ = ε/2K to verify the definition of equicontinuity of the sequence. This proves the following corollary:
- Let {ƒn} be a uniformly bounded sequence of real-valued differentiable functions on [a,b] such that the derivatives {ƒn′} is uniformly bounded. Then there exists a subsequence {ƒnk} that converges uniformly on [a,b].
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 ƒn are continuously differentiable with derivatives ƒn′. Suppose that ƒn′ are uniformly equicontinuous and uniformly bounded, and that the sequence ƒn is pointwise bounded (or just bounded at a single point). Then there is subsequence of the ƒn converging uniformly to a continuously differentiable function.
Lipschitz and Hölder continuous functions
The argument given above proves slightly more, specifically
- If {ƒn} is a uniformly bounded sequence of real valued functions on [a,b] such that each ƒ is Lipschitz continuous with the same Lipschitz constant K:
-
- for all x, y ∈ [a,b] and all ƒn, then there is a subsequence that converges uniformly on [a,b].
The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is
- A set F of functions ƒ on [a, b] that is uniformly bounded and satisfies a Hölder conditionHölder conditionIn mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...
of order α, 0 < α ≤ 1, with a fixed constant M,
- is relatively compact in C([a, b]). In particular, the unit ball of the Hölder spaceHölder conditionIn mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...
C 0, α([a, b]) is compact in C([a, b]).
This holds more generally for scalar functions on a compact metric space X satisfying a Hölder condition with respect to the metric on X.
Euclidean spaces
The Arzelà–Ascoli theorem holds, more generally, if the functions ƒn take values in d-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rd, and the proof is very simple: just apply the R-valued version of the Arzelà–Ascoli theorem d 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.
Proof
The proof is essentially based on a diagonalization argument. The simplest case is of real-valued functions on a closed and bounded interval:- Let be a closed and bounded interval. If F is an infinite set of functions ƒ : I → R which is uniformly bounded and equicontinuous, then there is a sequence ƒn of elements of F such that ƒn converges uniformly on I.
Fix an enumeration {xi}i=1,2,3,... of rational numbers in I. Since F is uniformly bounded, the set of points {ƒ(x1)}ƒ∈F is bounded, and hence by the Bolzano-Weierstrass theorem, there is a sequence {ƒn1} of distinct functions in F such that {ƒn1(x1)} converges. Repeating the same argument for the sequence of points {ƒn1(x2)}, there is a subsequence {ƒn2} of {ƒn1} such that {ƒn2(x2)} converges.
By mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
this process can be continued, and so there is a chain of subsequences
such that, for each k = 1, 2, 3, …, the subsequence {ƒnk} converges at x1,...,xk. Now form the diagonal subsequence {ƒ} whose mth term ƒm is the mth term in the mth subsequence {ƒnm}. By construction, ƒm converges at every rational point of I.
Therefore, given any ε > 0 and rational xk in I, there is an integer such that
Since the family F is equicontinuous, for this fixed ε and for every x in I, there is an open interval Ux containing x such that
for all ƒ ∈ F and all s, t in I such that s, t ∈ Ux.
The collection of intervals Ux, x ∈ I, forms an open cover of I. Since I is compact, this covering admits a finite subcover U1, ..., UJ. There exists an integer K such that each open interval Uj, 1 ≤ j ≤ J, contains a rational xk with 1 ≤ k ≤ K. Finally, for any t ∈ I, there are j and k so that t and xk belong to the same interval Uj. For this choice of k,
for all n, m > N = max{N(ε,x1), ..., N(ε,xK)}. Consequently, the sequence {ƒn} is uniformly Cauchy, and therefore converges to a continuous function, as claimed. This completes the proof.
Compact metric spaces and compact Hausdorff spaces
The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary compact metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s and, more generally still, compact Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s. Let X be a compact Hausdorff space, and let C(X) be the space of real-valued continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s on X. A subset is said to be equicontinuous if for every x ∈ X and every ε > 0, x has a neighborhood Ux such that
for all y ∈ Ux and ƒ ∈ F.
A set is said to be pointwise bounded if for every x ∈ X,
A version of this holds also in the space C(X) of real-valued continuous functions on a compact Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
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 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
Continuous functions on a compact Hausdorff space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by C, is a vector space with respect to the pointwise addition of functions...
.
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
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
with only minimal changes to the statement (see, for instance, ):
- Let X be a compact Hausdorff space and
a metric space. Then a subset F of C(X,Y) is compact in the compact-open topology 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.
The proof given 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 of any function in the family is less than ε on each open set 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.
Necessity
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. For each fixed x ∈ X and ε, sets of the formform an open covering of F, as U varies over open neighborhoods of x. Choosing a finite subcover then gives equicontinuity.
Examples
- To every function g that is p-integrable on [0, 1], 1 < p ≤ ∞, associate the function G defined on [0, 1] by
- Let F be the set of functions G corresponding to functions g in the unit ball of the space Lp([0, 1])Lp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. If q is the Hölder conjugate of p, defined by 1/p + 1/q = 1, then Hölder's inequality implies that all functions in F satisfy a Hölder condition with α = 1/q and constant M = 1.
- It follows that F is compact in C([0, 1]). This means that the correspondence g → G defines a compactCompact operatorIn functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
linear operator T between the Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s Lp([0, 1]) and C([0, 1]). Composing with the injection of C([0, 1]) into Lp([0, 1]), one sees that T acts compactly from Lp([0, 1]) to itself. The case p = 2 can be seen as a simple instance of the fact that the injection from the Sobolev spaceSobolev spaceIn mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
into L2(Ω), for Ω a bounded open set in Rd, is compact.
- When T is a compact linear operator from a Banach space X to a Banach space Y, its transpose T∗ is compact from the (continuous) dual Y∗ to X∗. This can be checked by the Arzelà–Ascoli theorem.
- Indeed, the image T(B) of the closed unit ball B of X is contained in a compact subset K of Y. The unit ball B∗ of Y∗ defines, by restricting from Y to K, a set F of (linear) continuous functions on K that is bounded and equicontinuous. By Arzelà–Ascoli, for every sequence {y∗n} in B∗, there is a subsequence that converges uniformly on K, and this implies that the image
of that subsequence is Cauchy in X∗.
- When ƒ is holomorphicHolomorphic functionIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
in an open disk D1 = B(z0, r), with modulus bounded by M, then (for example by Cauchy's formulaCauchy's integral formulaIn mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all...
) its derivative ƒ′ has modulus bounded by 4M ⁄ r in the smaller disk D2 = B(z0, r ⁄ 2). If a family of holomorphic functions on D1 is bounded by M on D1, it follows that the family F of restrictions to D2 is equicontinuous on D2. Therefore, a sequence converging uniformly on D2 can be extracted. This is a first step in the direction of Montel's theoremMontel's theoremIn complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after Paul Montel, and give conditions under which a family of holomorphic functions is normal....
.