Arzelà–Ascoli theorem

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 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 formula_1 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:
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:
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
The limit function is also Lipschitz continuous with the same value for the Lipschitz constant. A slight refinement is
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 , 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 spaces and, more generally still, compact Hausdorff spaces. Let "X" be a compact Hausdorff space, and let "C"("X") be the space of real-valued continuous functions on "X". A subset is said to be "equicontinuous" if for every "x" ∈ "X" and every , "x" has a neighborhood "Ux" such that
A set 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" :
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".
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. 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 formula_9 where formula_10 is not compact. Particularly important are cases where formula_10 is a topological vector space. Recall that if formula_10
is a topological space and formula_13 is a uniform space (such as any metric space or any topological group, metrisable or not), there is the topology of compact convergence on the set formula_14 of functions formula_9; 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 formula_10. Let formula_17 be the subspace of
formula_14 consisting of continuous functions, equipped with the topology of compact convergence.
Then one form of the Arzèla-Ascoli theorem is the following:
This theorem immediately gives the more specialised statements above in cases where formula_10 is compact
and the uniform structure of formula_13 is given by a metric. There are a few other variants in terms of
the topology of precompact convergence or other related topologies on 
formula_14. It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of formula_10 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 formula_31, 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 formula_32 the space of functions from formula_10 to formula_13 endowed with the uniform metric
Then we have the following:
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. Let "N"("ε", "U") be the set of all functions in F whose oscillation over an open subset "U" ⊂ "X" is less than "ε":
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.