Английская Википедия:Arzelà–Ascoli theorem

Шаблон:Short description 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 Шаблон:Harvtxt, who established the sufficient condition for compactness, and by Шаблон:Harvtxt, who established the necessary condition and gave the first clear presentation of the result. A further generalization of the theorem was proven by Шаблон:Harvtxt, to sets of real-valued continuous functions with domain a compact metric space Шаблон:Harv. 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 Шаблон:Harvtxt.

Statement and first consequences

By definition, a sequence <math>\{f_n\}_{n \in \mathbb{N}}</math> of continuous functions on an interval Шаблон:Math is uniformly bounded if there is a number Шаблон:Math such that

<math>\left|f_n(x)\right| \le M</math>

for every function Шаблон:Math belonging to the sequence, and every Шаблон:Math. (Here, Шаблон:Math must be independent of Шаблон:Math and Шаблон:Math.)

The sequence is said to be uniformly equicontinuous if, for every Шаблон:Math, there exists a Шаблон:Math such that

<math>\left|f_n(x)-f_n(y)\right| < \varepsilon</math>

whenever Шаблон:Math for all functions Шаблон:Math in the sequence. (Here, Шаблон:Math may depend on Шаблон:Math, but not Шаблон:Math, Шаблон:Math or Шаблон:Math.)

One version of the theorem can be stated as follows:

Consider a sequence of real-valued continuous functions Шаблон:Math defined on a closed and bounded interval Шаблон:Math of the real line. If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence Шаблон:Math that converges uniformly.

The converse is also true, in the sense that if every subsequence of Шаблон:Mathitself has a uniformly convergent subsequence, then Шаблон:Mathis uniformly bounded and equicontinuous.

Шаблон:Math proof

Immediate examples

Differentiable functions

The hypotheses of the theorem are satisfied by a uniformly bounded sequence Шаблон:Mathof differentiable functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the mean value theorem that for all Шаблон:Mvar and Шаблон:Mvar,

<math>\left|f_n(x) - f_n(y)\right| \le K |x-y|,</math>

where K is the supremum of the derivatives of functions in the sequence and is independent of Шаблон:Mvar. So, given Шаблон:Math, let Шаблон:Math to verify the definition of equicontinuity of the sequence. This proves the following corollary:

Let {f_n} be a uniformly bounded sequence of real-valued differentiable functions on Шаблон:Math such that the derivatives {f_n′} are uniformly bounded. Then there exists a subsequence {f_{n_k}} that converges uniformly on Шаблон:Math.

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 Шаблон:Math are continuously differentiable with derivatives Шаблон:Math. Suppose that f_n′ are uniformly equicontinuous and uniformly bounded, and that the sequence Шаблон:Math is pointwise bounded (or just bounded at a single point). Then there is a subsequence of the Шаблон:Mathconverging 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 Шаблон:Mathis a uniformly bounded sequence of real valued functions on Шаблон:Math such that each f is Lipschitz continuous with the same Lipschitz constant Шаблон:Mvar:

<math>\left|f_n(x) - f_n(y)\right| \le K|x-y|</math>

for all Шаблон:Math and all Шаблон:Math, then there is a subsequence that converges uniformly on Шаблон:Math.

The limit function is also Lipschitz continuous with the same value Шаблон:Mvar for the Lipschitz constant. A slight refinement is

A set Шаблон:Math of functions Шаблон:Math on Шаблон:Math that is uniformly bounded and satisfies a Hölder condition of order Шаблон:Math, Шаблон:Math, with a fixed constant Шаблон:Mvar,

<math>\left|f(x) - f(y)\right| \le M \, |x - y|^\alpha, \qquad x, y \in [a, b]</math>

is relatively compact in Шаблон:Math. In particular, the unit ball of the Hölder space Шаблон:Math is compact in Шаблон:Math.

This holds more generally for scalar functions on a compact metric space Шаблон:Mvar satisfying a Hölder condition with respect to the metric on Шаблон:Mvar.

Generalizations

Euclidean spaces

The Arzelà–Ascoli theorem holds, more generally, if the functions Шаблон:Math take values in Шаблон:Mvar-dimensional Euclidean space Шаблон:Math, and the proof is very simple: just apply the Шаблон:Math-valued version of the Arzelà–Ascoli theorem Шаблон:Mvar 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 Шаблон:Math is said to be equicontinuous if for every x ∈ X and every Шаблон:Math, x has a neighborhood U_x such that

<math>\forall y \in U_x, \forall f \in \mathbf{F} : \qquad |f(y) - f(x)| < \varepsilon.</math>

A set Шаблон:Math is said to be pointwise bounded if for every x ∈ X,

<math>\sup \{ | f(x) | : f \in \mathbf{F} \} < \infty.</math>

A version of the Theorem holds also in the space C(X) of real-valued continuous functions on a compact Hausdorff space X Шаблон:Harv:

Let X be a compact Hausdorff space. Then a subset F of C(X) is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and pointwise bounded.

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, Шаблон:Harvtxt, Шаблон:Harvtxt):

Let X be a compact Hausdorff space and Y a metric space. Then Шаблон:Math is compact in the compact-open topology if and only if it is equicontinuous, pointwise relatively compact and closed.

Here pointwise relatively compact means that for each x ∈ X, the set Шаблон:Mathis 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 <math>X \rightarrow Y</math> where <math>X</math> is not compact. Particularly important are cases where <math>X</math> is a topological vector space. Recall that if <math>X</math> is a topological space and <math>Y</math> 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 <math>\mathfrak{F}(X,Y)</math> of functions <math>X \rightarrow Y</math>; 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 <math>X</math>. Let <math>\mathcal{C}_c(X,Y)</math> be the subspace of <math>\mathfrak{F}(X,Y)</math> consisting of continuous functions, equipped with the topology of compact convergence. Then one form of the Arzèla-Ascoli theorem is the following:

Let <math>X</math> be a topological space, <math>Y</math> a Hausdorff uniform space and <math>H\subset\mathcal{C}_c(X,Y)</math> an equicontinuous set of continuous functions such that <math>H(x)</math> is relatively compact in <math>Y</math> for each <math>x\in X</math>. Then <math>H</math> is relatively compact in <math>H\subset\mathcal{C}_c(X,Y)</math>.

This theorem immediately gives the more specialised statements above in cases where <math>X</math> is compact and the uniform structure of <math>Y</math> is given by a metric. There are a few other variants in terms of the topology of precompact convergence or other related topologies on <math>\mathfrak{F}(X,Y)</math>. It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of <math>X</math> 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 <math>0</math>, 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. Шаблон:Harvtxt).

Denote by <math>S(X,Y)</math> the space of functions from <math>X</math> to <math>Y</math> endowed with the uniform metric

Then we have the following:

Let <math>X</math> be a compact metric space and <math>Y</math> a complete metric space. Let <math>\{v_n\}_{n\in\mathbb{N}}</math> be a sequence in <math>S(X,Y)</math> such that there exists a function <math>\omega:X\times X\to[0,\infty]</math> and a sequence <math>\{\delta_n\}_{n\in\mathbb{N}}\subset[0,\infty)</math> satisfying

<math>\lim_{d_X(t,t')\to0}\omega(t,t')=0,\quad\lim_{n\to\infty}\delta_n=0,</math>

<math>\forall(t,t')\in X\times X,\quad \forall n\in\mathbb{N},\quad d_Y(v_n(t),v_n(t'))\leq \omega(t,t')+\delta_n.</math>

Assume also that, for all <math>t\in X</math>, <math>\{v_n(t):n\in\mathbb{N}\}</math> is relatively compact in <math>Y</math>. Then <math>\{v_n\}_{n\in\mathbb{N}}</math> is relatively compact in <math>S(X,Y)</math>, and any limit of <math>\{v_n\}_{n\in\mathbb{N}}</math> in this space is in <math>C(X,Y)</math>.

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:

<math>N(\varepsilon, U) = \{f \mid \operatorname{osc}_U f < \varepsilon\}.</math>

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.

Further examples

To every function Шаблон:Mvar that is [[Lp space#Lp spaces and Lebesgue integrals|Шаблон:Mvar-integrable]] on Шаблон:Math, with Шаблон:Math, associate the function Шаблон:Mvar defined on Шаблон:Math by

<math>G(x) = \int_0^x g(t) \, \mathrm{d}t.</math>

Let Шаблон:Math be the set of functions Шаблон:Mvar corresponding to functions Шаблон:Mvar in the unit ball of the space Шаблон:Math. If Шаблон:Mvar is the Hölder conjugate of Шаблон:Mvar, defined by Шаблон:Math, then Hölder's inequality implies that all functions in Шаблон:Math satisfy a Hölder condition with Шаблон:Math and constant Шаблон:Math.

It follows that Шаблон:Math is compact in Шаблон:Math. This means that the correspondence Шаблон:Math defines a compact linear operator Шаблон:Mvar between the Banach spaces Шаблон:Math and Шаблон:Math. Composing with the injection of Шаблон:Math into Шаблон:Math, one sees that Шаблон:Mvar acts compactly from Шаблон:Math to itself. The case Шаблон:Math can be seen as a simple instance of the fact that the injection from the Sobolev space <math>H^1_0(\Omega)</math> into Шаблон:Math, for Шаблон:Math a bounded open set in Шаблон:Math, is compact.

When Шаблон:Mvar is a compact linear operator from a Banach space Шаблон:Mvar to a Banach space Шаблон:Mvar, its transpose Шаблон:Math is compact from the (continuous) dual Шаблон:Math to Шаблон:Math. This can be checked by the Arzelà–Ascoli theorem.

Indeed, the image Шаблон:Math of the closed unit ball Шаблон:Mvar of Шаблон:Mvar is contained in a compact subset Шаблон:Mvar of Шаблон:Mvar. The unit ball Шаблон:Math of Шаблон:Math defines, by restricting from Шаблон:Mvar to Шаблон:Mvar, a set Шаблон:Math of (linear) continuous functions on Шаблон:Mvar that is bounded and equicontinuous. By Arzelà–Ascoli, for every sequence Шаблон:Math in Шаблон:Math, there is a subsequence that converges uniformly on Шаблон:Mvar, and this implies that the image <math>T^*(y^*_{n_k})</math> of that subsequence is Cauchy in Шаблон:Math.

When Шаблон:Math is holomorphic in an open disk Шаблон:Math, with modulus bounded by Шаблон:Mvar, then (for example by Cauchy's formula) its derivative Шаблон:Math has modulus bounded by Шаблон:Math in the smaller disk Шаблон:Math If a family of holomorphic functions on Шаблон:Math is bounded by Шаблон:Mvar on Шаблон:Math, it follows that the family Шаблон:Math of restrictions to Шаблон:Math is equicontinuous on Шаблон:Math. Therefore, a sequence converging uniformly on Шаблон:Math can be extracted. This is a first step in the direction of Montel's theorem.
Let <math>C([0,T],L^1(\mathbb{R}^N))</math> be endowed with the uniform metric <math>\textstyle\sup_{t\in [0,T]}\|v(\cdot,t)-w(\cdot,t)\|_{L^1(\mathbb{R}^N)}.</math> Assume that <math>u_n=u_n(x,t)\subset C([0,T];L^1(\mathbb{R}^N))</math> is a sequence of solutions of a certain partial differential equation (PDE), where the PDE ensures the following a priori estimates: <math>x\mapsto u_n(x,t)</math> is equicontinuous for all <math>t</math>, <math>x\mapsto u_n(x,t)</math> is equitight for all <math>t</math>, and, for all <math>(t,t')\in [0,T]\times[0,T]</math> and all <math>n\in\mathbb{N}</math>, <math>\|u_n(\cdot,t)-u_n(\cdot,t')\|_{L^1(\mathbb{R}^N)}</math> is small enough when <math>|t-t'|</math> is small enough. Then by the Fréchet–Kolmogorov theorem, we can conclude that <math>\{x\mapsto u_n(x,t):n\in\mathbb{N}\}</math> is relatively compact in <math>L^1(\mathbb{R}^N)</math>. Hence, we can, by (a generalization of) the Arzelà–Ascoli theorem, conclude that <math>\{u_n:n\in\mathbb{N}\}</math> is relatively compact in <math>C([0,T],L^1(\mathbb{R}^N)).</math>

References

Шаблон:Reflist Шаблон:Reflist

Шаблон:Functional analysis Шаблон:PlanetMath attribution

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Английская Википедия:Arzelà–Ascoli theorem

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