Arzelà–Ascoli theorem

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

Statement and first consequences

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

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

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

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

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

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

One version of the theorem can be stated as follows:

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

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

Immediate examples

Differentiable functions

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

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

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

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

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

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

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

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

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

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

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

Generalizations

Euclidean spaces

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

Let X be a compact Hausdorff space and Y a metric space. Then Template: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 Template: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 Arzelà-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) : h \in H\}</math> is relatively compact in <math>Y</math> for each <math>x\in X</math>. Then <math>H</math> is relatively compact in <math>\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. Template: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

<math>d_S(v,w)=\sup_{t\in X}d_Y(v(t),w(t)).</math>

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

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

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

It follows that Template:Math is compact in Template:Math. This means that the correspondence Template:Math defines a compact linear operator Template:Mvar between the Banach spaces Template:Math and Template:Math. Composing with the injection of Template:Math into Template:Math, one sees that Template:Mvar acts compactly from Template:Math to itself. The case Template: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 Template:Math, for Template:Math a bounded open set in Template:Math, is compact.

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

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

• When Template:Math is holomorphic in an open disk Template:Math, with modulus bounded by Template:Mvar, then (for example by Cauchy's formula) its derivative Template:Math has modulus bounded by Template:Math in the smaller disk Template:Math If a family of holomorphic functions on Template:Math is bounded by Template:Mvar on Template:Math, it follows that the family Template:Math of restrictions to Template:Math is equicontinuous on Template:Math. Therefore, a sequence converging uniformly on Template: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>

See also

• Helly's selection theorem
• Fréchet–Kolmogorov theorem

References

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• Arzelà-Ascoli theorem at Encyclopaedia of Mathematics
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