Arzelà–Ascoli theorem

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 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).

Contents 1 Statement and first consequences 1.1 Immediate examples 1.1.1 Differentiable functions 1.1.2 Lipschitz and Hölder continuous functions 2 Generalizations 2.1 Euclidean spaces 2.2 Compact metric spaces and compact Hausdorff spaces 2.3 Functions on non-compact spaces 2.4 Non-continuous functions 3 Necessity 4 Further examples 5 See also 6 References Statement and first consequences By definition, a sequence {displaystyle {f_{n}}_{nin mathbb {N} }} of continuous functions on an interval I = [a, b] is uniformly bounded if there is a number M such that {displaystyle left|f_{n}(x)right|leq M} for every function  fn  belonging to the sequence, and every x ∈ [a, b]. (Here, M must be independent of n and x.) The sequence is said to be uniformly equicontinuous if, for every ε > 0, there exists a δ > 0 such that {displaystyle left|f_{n}(x)-f_{n}(y)right| 0 and rational xk in I, there is an integer N = N(ε, xk) such that {displaystyle |f_{n}(x_{k})-f_{m}(x_{k})|<{tfrac {varepsilon }{3}},qquad n,mgeq 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 {displaystyle |f(s)-f(t)|<{tfrac {varepsilon }{3}}} for all f ∈ 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 closed and bounded, by the Heine-Borel theorem I is compact, implying that 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, {displaystyle {begin{aligned}left|f_{n}(t)-f_{m}(t)right|&leq left|f_{n}(t)-f_{n}(x_{k})right|+|f_{n}(x_{k})-f_{m}(x_{k})|+|f_{m}(x_{k})-f_{m}(t)|\&<{tfrac {varepsilon }{3}}+{tfrac {varepsilon }{3}}+{tfrac {varepsilon }{3}}end{aligned}}} for all n, m > N = max{N(ε, x1), ..., N(ε, xK)}. Consequently, the sequence {fn} is uniformly Cauchy, and therefore converges to a continuous function, as claimed. This completes the proof.

Immediate examples Differentiable functions The hypotheses of the theorem are satisfied by a uniformly bounded sequence { fn } of differentiable functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the mean value theorem that for all x and y, {displaystyle left|f_{n}(x)-f_{n}(y)right|leq K|x-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. This proves the following corollary: Let {fn} be a uniformly bounded sequence of real-valued differentiable functions on [a, b] such that the derivatives {fn′} are uniformly bounded. Then there exists a subsequence {fnk} that converges uniformly on [a, b].

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  fn  are continuously differentiable with derivatives f′n. 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 a subsequence of the { fn } 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 { fn } is a uniformly bounded sequence of real valued functions on [a, b] such that each f is Lipschitz continuous with the same Lipschitz constant K: {displaystyle left|f_{n}(x)-f_{n}(y)right|leq K|x-y|} for all x, y ∈ [a, b] and all  fn , then there is a subsequence that converges uniformly on [a, b].

The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is A set F of functions  f  on [a, b] that is uniformly bounded and satisfies a Hölder condition of order α, 0 < α ≤ 1, with a fixed constant M, {displaystyle left|f(x)-f(y)right|leq M,|x-y|^{alpha },qquad x,yin [a,b]} is relatively compact in C([a, b]). In particular, the unit ball of the Hölder space C0,α([a, b]) is compact in C([a, b]). This holds more generally for scalar functions on a compact metric space X satisfying a Hölder condition with respect to the metric on X. Generalizations Euclidean spaces 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. 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 F ⊂ C(X) is said to be equicontinuous if for every x ∈ X and every ε > 0, x has a neighborhood Ux such that {displaystyle forall yin U_{x},forall fin mathbf {F} :qquad |f(y)-f(x)|

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