Arzelà–Ascoli theorem

Generalizations

Euclidean spaces

The Arzelà–Ascoli theorem holds, more generally, if the functions f_n take values in d-dimensional Euclidean space R^d, 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.

Locally compact Hausdorff spaces

The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary space X. Let C(X, Y) be the space of all continuous functions from X to Y. If Y is a metric space with metric d, then a subset F ⊂ C(X, Y) is said to be equicontinuous if for every x ∈ X and every ε > 0, x has a neighborhood U_x such that

${\displaystyle \mathrm{\forall }y\in U_x,\mathrm{\forall }f\in \mathbf{F}:d(f(y),f(x))<\epsilon .}$

A set F ⊂ C(X, R) is said to be pointwise bounded if for every x ∈ X,

${\displaystyle sup\{|f(x)|:f\in \mathbf{F}\}<\mathrm{\infty }.}$

A version of the theorem holds also in the space C(X) of real-valued continuous functions on a compact Hausdorff space X (Dunford & Schwartz 1958, §IV.6.7):

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 domain can be locally compact or compactly generated, and the functions can assume values in a metric space or (Hausdorff) uniform space with only minimal changes to the statement (see, for instance, Kelley & Namioka (1982, §8), Kelley (1991, Chapter 7)):

Let X be a locally compact Hausdorff space and Y a metric space. Then F ⊂ C(X, Y) is relatively compact in the compact-open topology if and only if it is equicontinuous and pointwise relatively compact.

Here pointwise relatively compact means that for each x ∈ X, the set F_x = {f(x) : f ∈ F} is 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 Arzelà-Ascoli theorem generalises to functions ${\displaystyle X\to Y}$ where ${\displaystyle X}$ is not compact. Particularly important are cases where ${\displaystyle X}$ is a topological vector space. Recall that if ${\displaystyle X}$ is a topological space and ${\displaystyle Y}$ 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 ${\displaystyle \mathfrak{F}(X,Y)}$ of functions ${\displaystyle X\to Y}$; 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 ${\displaystyle X}$. Let ${\displaystyle {\mathcal{C}}_c(X,Y)}$ be the subspace of ${\displaystyle \mathfrak{F}(X,Y)}$ consisting of continuous functions, equipped with the topology of compact convergence. Then one form of the Arzelà-Ascoli theorem is the following:

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

This theorem immediately gives the more specialised statements above in cases where ${\displaystyle X}$ is compact and the uniform structure of ${\displaystyle Y}$ is given by a metric. There are a few other variants in terms of the topology of precompact convergence or other related topologies on ${\displaystyle \mathfrak{F}(X,Y)}$. It is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of ${\displaystyle X}$ 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 ${\displaystyle 0}$, 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. Droniou & Eymard (2016, Appendix)).

Denote by ${\displaystyle S(X,Y)}$ the space of functions from ${\displaystyle X}$ to ${\displaystyle Y}$ endowed with the uniform metric

${\displaystyle d_S(v,w)=\underset{t\in X}{sup}{d}_Y(v(t),w(t)).}$

Then we have the following:

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

${\displaystyle \underset{d_X(t,t^{\prime })\to 0}{lim}\omega (t,t^{\prime })=0,\underset{n\to \mathrm{\infty }}{lim}{\delta }_n=0,}$

${\displaystyle \mathrm{\forall }(t,t^{\prime })\in X\times X,\mathrm{\forall }n\in \mathbb{N},d_Y(v_n(t),v_n(t^{\prime }))\le \omega (t,t^{\prime })+{\delta }_n.}$

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

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:

${\displaystyle N(\epsilon ,U)=\{f\mid {\mathrm{osc}}_{U}f<\epsilon \}.}$

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 g that is p-integrable on [0, 1], with 1 < p ≤ ∞, associate the function G defined on [0, 1] by

${\displaystyle G(x)={\int }_0^{x}g(t)\mathrm{d}t.}$

Let F be the set of functions G corresponding to functions g in the unit ball of the space L^p([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 L^p([0, 1]) and C([0, 1]). Composing with the injection of C([0, 1]) into L^p([0, 1]), one sees that T acts compactly from L^p([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 ${\displaystyle H_0^1(\mathrm{\Omega })}$ into L²(Ω), for Ω a bounded open set in R^d, 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 ${\displaystyle T^{\ast }(y_{n_k}^{\ast })}$ of that subsequence is Cauchy in X^∗.

• When f is holomorphic in an open disk D₁ = B(z₀, r), with modulus bounded by M, then (for example by Cauchy's formula) its derivative f′ has modulus bounded by ⁠2M/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.
• Let ${\displaystyle C([0,T],L^1({\mathbb{R}}^N))}$ be endowed with the uniform metric ${\displaystyle \underset{t\in [0,T]}{sup}\Vert v(\cdot ,t)-w(\cdot ,t){\Vert }_{L^1({\mathbb{R}}^N)}.}$ Assume that ${\displaystyle u_n=u_n(x,t)\subset C([0,T];L^1({\mathbb{R}}^N))}$ is a sequence of solutions of a certain partial differential equation (PDE), where the PDE ensures the following a priori estimates: ${\displaystyle x\mapsto u_n(x,t)}$ is equicontinuous for all ${\displaystyle t}$, ${\displaystyle x\mapsto u_n(x,t)}$ is equitight for all ${\displaystyle t}$, and, for all ${\displaystyle (t,t^{\prime })\in [0,T]\times [0,T]}$ and all ${\displaystyle n\in \mathbb{N}}$, ${\displaystyle \Vert u_n(\cdot ,t)-u_n(\cdot ,t^{\prime }){\Vert }_{L^1({\mathbb{R}}^N)}}$ is small enough when ${\displaystyle |t-t^{\prime }|}$ is small enough. Then by the Fréchet–Kolmogorov theorem, we can conclude that ${\displaystyle \{x\mapsto u_n(x,t):n\in \mathbb{N}\}}$ is relatively compact in ${\displaystyle L^1({\mathbb{R}}^N)}$. Hence, we can, by (a generalization of) the Arzelà–Ascoli theorem, conclude that ${\displaystyle \{u_n:n\in \mathbb{N}\}}$ is relatively compact in ${\displaystyle C([0,T],L^1({\mathbb{R}}^N)).}$