Ascoli-Arzelà theorem
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In mathematics, the Arzelà–Ascoli theorem of functional analysis gives necessary and sufficient conditions to decide whether a set of continuous functions from a compact metric space into a metric space is compact in the topology of uniform convergence. The main condition is equicontinuity of the set of functions.

The notion of equicontinuity was introduced at around the same time by Ascoli (1883–1884) and Arzelà (1882–1883). 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) for any space in which the notion of a limit makes sense (such as a metric space or Hausdorff space).

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.

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Arzelà–Ascoli theorem

Real line

In simplest terms, the theorem can be stated as follows :

Consider a sequence of continuous functions (fn)n∈N defined on a closed and bounded interval a, b of the real line with real values. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence (fnk) that converges uniformly.

For example, the theorem's hypotheses are satisfied by a uniformly bounded sequence of differentiable functions with uniformly bounded derivatives. If the sequence of second derivatives is also uniformly bounded, then the derivatives also converge uniformly, and so on.

Another generalization holds for continuously differentiable functions. Suppose that the functions fn are continuously differentiable with derivatives fn′. Suppose that fn′ are uniformly equicontinuous and uniformly bounded, and that the sequence fn is pointwise bounded (or just bounded at a single point). Then there is subsequence of the fn converging uniformly to a continuously differentiable function.

The above theorem also holds 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. More generally, the theorem holds for mappings between two Euclidean spaces.

Metric space

The general version of this theorem for metric spaces is as follows :

Let X be a compact metric space, Y 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, C(X,Y) denotes the set of all continuous functions from X to Y, and a subset F is pointwise relatively compact if and only if \forall x \in X, the set {f(x):f is in F} is relatively compact in Y. If a set is compact in the compact-open topology, then in particular every sequence from the set has a subsequence which converges uniformly on compact subsets.

More generally, this holds if X is a compact Hausdorff space (Dunford & Schwartz 1958, §IV.6.7):

Let X be a compact Hausdorff space. Then a subset F of C(X) is compact in the compact-open topology if and only if it is equicontinuous, pointwise relatively compact and closed.

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, Kelley & Namioka (1982, §8), Kelley (1975, Chapter 7)):

Let X be a compact Hausdorff space and Y 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.

Proof of a version of Arzelà–Ascoli theorem

We will prove here the following version of the theorem, valid for real-valued functions on closed and bounded intervals in R. Proofs of other versions of the theorem are quite similar, provided the necessary parts of the proof are abstracted to the more general situation.

  • Let I ⊂ R be a closed and bounded interval. If F = {ƒ} 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 this process can be continued, and so there is a chain of subsequences

\{f_{n_1}\}\supset \{f_{n_2}\}\supset \cdots

such that, for each k = 1, 2, 3, …, the subsequence {ƒnk} converges at x1,...,xk. Now form the diagonal subsequence {fm} whose mth term fm is the mth term in the mth subsequence \{f_{n_m}\}. By construction, ƒm converges at every rational point of I.

Therefore, given any ε > 0 and rational xk in I, there is an integer N = N(ε,xk) such that

|f_n(x_k) - f_m(x_k)| < \varepsilon/3,\quad n,m \ge N.\,

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

|f(s)-f(t)| < \varepsilon/3\,

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,

\begin{align} |f_n(t)-f_m(t)| & {} \le |f_n(t) - f_n(x_k)| + |f_n(x_k) - f_m(x_k)| + |f_m(x_k) - f_m(t)| \\ & {} < \varepsilon/3 + \varepsilon/3 +\varepsilon/3 \end{align}

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.

Examples

  • The set F of functions ƒ on [0, 1] that are bounded by 1 and satisfy a Hölder condition of order α, 0 < α ≤ 1, with a fixed constant M,
|f(x) - f(y)] \le M \, |x - y|^\alpha, \quad x, y \in [0, 1]
is compact in C([0, 1]). In other words, the unit ball of the Hölder space  C0,α([0,1]) is compact in C([0, 1]).
This holds more generally for scalar functions on a compact metric space X satisfying a Hölder condition with respect to the metric on X.
  • To every function g that is p-integrable on [0, 1], 1 < p ≤ ∞, associate the function G defined on [0, 1] by
G(x) = \int_0^x g(t) \, \mathrm{d}t.
Let F be the set of functions G corresponding to functions g in the unit ball of the space Lp([0, 1]). 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 compact linear operator T  between the Banach spaces 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 space H^1_0(\Omega) 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 T^*(y^*_{n_k})  of that subsequence is Cauchy in X∗.
  • When ƒ is holomorphic in a disk D1 = B(z0, r), with modulus bounded by M, then (for example by Cauchy's formula) 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 theorem.

References

  • Arzelà, Cesare (1895), "Sulle funzioni di linee", Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5): 55–74 .
  • Arzelà, Cesare (1882–1883), "Un'osservazione intorno alle serie di funzioni", Rend. dell' Accad. R. delle Sci. dell'Istituto di Bologna: 142–159 .
  • Ascoli, G. (1883–1884), "Le curve limiti di una varietà data di curve", Atti della R. Accad. dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. 18 (3): 521–586 .
  • Bourbaki, Nicolas (1966), Elements of mathematics, General topology, Hermann .
  • Dieudonné, Jean (1988), Foundations of modern analysis, Academic Press, ISBN 978-0122155079 
  • Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume 1, Wiley-Interscience .
  • Fréchet, Maurice (1906), "Sur quelques points du calcul fonctionnel", Rend. Circ. Mat. Palermo 22: 1–74 .
  • Kelley, J. L. (1975), General topology, Springer-Verlag, ISBN 978-0387901251 
  • Kelley, J. L.; Namioka, I. (1982), Linear Topological Spaces, Springer-Verlag, ISBN 978-0387901695 
  • Rudin, Walter (1976), Principles of mathematical analysis, McGraw-Hill, ISBN 978-0070542358 

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