Arzelà–Ascoli 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.

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[edit] Arzelà–Ascoli theorem

[edit] 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 uniformly 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.

The above theorem also holds if the functions fn take values in n-dimensional Euclidean space Rn, and the proof is very simple: just apply the R-valued version of the Arzelà–Ascoli theorem n 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.

[edit] 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 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.

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

Let X be a compact Hausdorff space and Y a metric space. Then a subset F of C(X,Y) is compact if and only if it is equicontinuous, pointwise relatively compact and closed.

The Arzelà–Ascoli theorem is a fundamental result in the study of the algebra of continuous functions on a compact Hausdorff space.

[edit] 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.

To prove this, 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 by Zorn's lemma 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 sequence

f_n = f_{n_n}.\,

By construction, ƒn converges at every rational point of I.

Therefore, given any ε > 0 and rational xk in I, there is an integer N = N(ε/3,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, there exists a δ > 0 such that

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

for all ƒ ∈ F and all s, t ∈ I such that |s − t| < δ.

Consider the collection of intervals centered at rational points in I of length less than δ. This forms an open cover of I. Since I is compact, this covering admits a finite subcover I1, ..., Ik, with respective rational centers x1, ..., xn. Finally, for any t ∈ I, there is a k so that t ∈ Ik. 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/3, x1), ..., N(ε/3,xk)}. Consequently, the sequence {ƒn} is uniformly Cauchy, and therefore converges to a continuous function, as claimed.

[edit] 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 mathemetics, General topology, Hermann .
  • Fréchet, Maurice (1906), “Sur quelques points du calcul fonctionnel”, Rend. Circ. Mat. Palermo 22: 1–74 .
  • Rudin, Walter (1976), Principles of mathematical analysis, McGraw-Hill, ISBN 978-0070542358 
  • Dieudonné, Jean (1988), Foundations of modern analysis, Academic Press, ISBN 978-0122155079 
  • Dunford, Nelson & Schwartz, Jacob T. (1958), Linear operators, volume 1, Wiley-Interscience .

This article incorporates material from Ascoli–Arzelà theorem on PlanetMath, which is licensed under the GFDL.