The à–Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. 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, Montel's theorem in complex analysis, 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 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; see .
Statement and first consequences
By definition, a sequence of continuous functions 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 of real-valued continuous functions defined on a closed and bounded interval of the real line. If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence 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 that for all and ,
where K is the supremum 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 { fn} be a uniformly bounded sequence of real-valued differentiable functions on such that the derivatives { fn′} are uniformly bounded. Then there exists a subsequence { fnk} 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 fn′ 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 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 of order , , with a fixed constant ,
is relatively compact in . In particular, the unit ball of the Hölder space is compact in .