The 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 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 Ascoli (1883–1884), who established the sufficient condition for compactness, and by Arzelà (1895), who established the necessary condition and gave the first clear presentation of the result.
A further generalization of the theorem was proven by Fréchet (1906), to sets of real-valued continuous functions with domain a compact metric space (Dunford & Schwartz 1958, p. 382).
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 Kelley (1991, page 234).
(Here, δ may depend on ε, but not x, y or n.) One version of the theorem can be stated as follows: The proof is essentially based on a diagonalization argument.
The simplest case is of real-valued functions on a closed and bounded interval: Fix an enumeration {xi}i ∈N of rational numbers in I.
Since I is closed and bounded, by the Heine–Borel theorem I is compact, implying that this covering admits a finite subcover U1, ..., UJ.
Consequently, the sequence {fn} is uniformly Cauchy, and therefore converges to a continuous function, as claimed.
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.
Then there is a subsequence of the { fn } converging uniformly to a continuously differentiable function.
The Arzelà–Ascoli theorem holds, more generally, if the functions fn 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.
A subset F ⊂ C(X) is said to be equicontinuous if for every x ∈ X and every ε > 0, x has a neighborhood Ux such that A set F ⊂ C(X, R) is said to be pointwise bounded if for every x ∈ X, A version of the Theorem holds also in the space C(X) of real-valued continuous functions on a compact Hausdorff space X (Dunford & Schwartz 1958, §IV.6.7): The Arzelà–Ascoli theorem is thus a fundamental result in the study of the algebra of continuous functions on a compact Hausdorff space.
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, Kelley & Namioka (1982, §8), Kelley (1991, Chapter 7)): Here pointwise relatively compact means that for each x ∈ X, the set Fx = { f (x) : f ∈ F} 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 of any function in the family is less than ε on each open set in the cover.
A similar argument is used as a part of the proof for the general version which does not assume completeness of Y.
consisting of continuous functions, equipped with the topology of compact convergence.
It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of
Solutions of numerical schemes for parabolic equations are usually piecewise constant, and therefore not continuous, in time.
, 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. Droniou & Eymard (2016, Appendix)).
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.