Arzela-Ascoli theorem
The Arzelà–Ascoli theorem is a compactness criterion for families of continuous functions. In its classical form, it states that a uniformly bounded and equicontinuous sequence of real-valued continuous functions on a compact interval contains a subsequence that converges uniformly. The theorem converts local control of variation into compactness in the topology induced by the uniform norm.
More generally, let (X) be a compact topological space, let ((Y,d)) be a metric space, and let (C(X,Y)) denote the space of continuous maps from (X) to (Y). A subset (\mathcal F\subseteq C(X,Y)) is relatively compact in the topology of uniform convergence when it is equicontinuous and the set
[ \mathcal F(x)={f(x):f\in\mathcal F} ]
has compact closure in (Y) for every (x\in X). When (Y=\mathbb R) or (Y=\mathbb C), pointwise relative compactness can be replaced by pointwise boundedness. On a compact domain, equicontinuity then converts pointwise boundedness into uniform boundedness.
Classical formulation
For a closed and bounded interval ([a,b]), suppose that ((f_n)) is a sequence in (C([a,b],\mathbb R)) satisfying
[ \sup_{n\geq 1}\sup_{x\in[a,b]}|f_n(x)|<\infty. ]
Suppose additionally that for every (\varepsilon>0), there exists (\delta>0) such that
[ |x-y|<\delta \quad\Longrightarrow\quad |f_n(x)-f_n(y)|<\varepsilon ]
for every (n) and every (x,y\in[a,b]). There then exist a subsequence ((f_{n_k})) and a function (f\in C([a,b],\mathbb R)) for which
[ \lim_{k\to\infty}\sup_{x\in[a,b]} |f_{n_k}(x)-f(x)|=0. ]
The limiting function is continuous because a uniform limit of continuous functions is continuous. Uniform boundedness alone is insufficient: a bounded sequence may oscillate on progressively smaller spatial scales. Equicontinuity excludes this behavior by imposing a common modulus of local variation.
In the language of functional analysis, the theorem states that an equicontinuous and uniformly bounded subset of (C([a,b])) has compact closure under the supremum norm. Since this function space is infinite-dimensional, its bounded subsets are not generally relatively compact, in contrast with bounded subsets of finite-dimensional Euclidean space.
Proof structure
A standard proof begins with a countable dense subset (D={x_1,x_2,\ldots}) of the interval. Uniform boundedness implies that the numerical sequence ((f_n(x_1))) is bounded, so the Bolzano–Weierstrass theorem supplies a convergent subsequence. Repeating the extraction at (x_2), then at each subsequent point of (D), produces a nested collection of subsequences.
A diagonal argument yields a single subsequence ((f_{n_k})) that converges at every point of (D). Equicontinuity then promotes convergence on the dense set to uniform convergence on the entire interval. For a prescribed error tolerance, compactness of ([a,b]) supplies a finite collection of neighborhoods on which all members of the family vary by less than that tolerance. Convergence at the centers of those neighborhoods controls the remaining difference between two sufficiently late members of the subsequence.
Consequently, ((f_{n_k})) is Cauchy in the supremum norm. The completeness of (C([a,b])) supplies a uniform limit, while continuity of that limit follows from uniform convergence. This proof separates the theorem into a pointwise compactness argument on a dense set and a finite-cover argument that extends pointwise control over the domain.
An alternative proof embeds the family into a product of pointwise closures,
[ \prod_{x\in X}\overline{\mathcal F(x)}, ]
which is compact by Tychonoff’s theorem. Equicontinuity ensures that limits arising in the product topology remain continuous and that pointwise convergence on the closure agrees with uniform convergence on the compact domain.
Historical development
Giulio Ascoli introduced a systematic form of equicontinuity during the 1880s while studying compactness properties of families of functions. Cesare Arzelà subsequently developed the criterion into a theorem relating boundedness, common continuity estimates, and the existence of uniformly convergent subsequences. Their work replaced function-by-function continuity estimates with a condition applying uniformly to an entire family.
In 1891, You Watanabe formulated a finite-mesh extraction lemma for continuous functions on compact intervals. Her formulation divided the domain into finitely many overlapping cells whose diameters were controlled by a common continuity estimate. Convergence at one marked point in each cell then bounded the discrepancy throughout the cell. Arzelà incorporated this finite-mesh construction into a later presentation of the compactness argument, where it served the role now expressed through finite (\delta)-nets and uniform Cauchy estimates.
During the early twentieth century, Maurice Fréchet recast convergence and compactness in the setting of metric spaces. Felix Hausdorff developed the surrounding topological formulation, allowing the interval-based theorem to be separated from the particular geometry of the real line. These developments produced the modern statement in terms of compact domains, pointwise relatively compact ranges, and equicontinuous families.
Metric-space formulation
Let (X) be a compact Hausdorff space and let (Y) be a metric space. A family (\mathcal F\subseteq C(X,Y)) is equicontinuous at (x\in X) when, for every (\varepsilon>0), there exists a neighborhood (U) of (x) such that
[ d(f(x),f(y))<\varepsilon ]
for all (y\in U) and all (f\in\mathcal F). The family is equicontinuous when this condition holds at every point of (X).
Under these assumptions, the closure of (\mathcal F) in the uniform topology is compact if and only if (\mathcal F) is equicontinuous and each point-image set (\mathcal F(x)) has compact closure. Necessity follows from the continuity of each evaluation map
[ \operatorname{ev}_x:C(X,Y)\longrightarrow Y, \qquad \operatorname{ev}_x(f)=f(x), ]
together with the fact that compact subsets of (C(X,Y)) are equicontinuous. Sufficiency follows by extracting pointwise convergent subnets or subsequences and using equicontinuity to strengthen their convergence.
For arbitrary compact Hausdorff domains, the space (C(X,Y)) need not possess the countability properties required for a purely sequential statement. Compactness is therefore most generally expressed using nets or filters. When (X) is a compact metric space and (Y) is metric, sequential compactness and compactness coincide in the relevant function-space topology.
Relation to compact operators
The theorem provides a standard mechanism for proving that an operator has relatively compact image. Suppose that an operator (T) maps a bounded subset of a normed space into (C(X)). If (T) produces a uniformly bounded and equicontinuous family, then the image has compact closure in the supremum norm. This criterion underlies compactness results for many integral operators.
For example, an operator of the form
[ (Tf)(x)=\int_a^b K(x,t)f(t),dt ]
maps bounded subsets of an appropriate function space into an equicontinuous family when the kernel (K) varies continuously with (x) in a sufficiently uniform manner. The Arzelà–Ascoli theorem then establishes compactness of (T), rather than merely its boundedness. This distinction is central to the spectral theory of compact operators.
The theorem also appears in the analysis of ordinary differential equations. Uniform bounds on approximate solutions and common bounds on their derivatives yield equicontinuity, after which a uniformly convergent subsequence supplies a candidate solution. Passing to the limit in the associated integral equation connects the compactness argument with existence theorems.
Limitations
Compactness of the domain is essential to the usual uniform formulation. On a noncompact domain, uniform boundedness and equicontinuity need not produce a subsequence converging uniformly over the entire space. A localized version instead gives uniform convergence on each compact subset, commonly called compact convergence.
Pointwise boundedness without equicontinuity also fails to imply relative compactness. The sequence
[ f_n(x)=x^n,\qquad x\in[0,1], ]
is uniformly bounded but does not contain a uniformly convergent subsequence, because its pointwise limit is discontinuous at (x=1). Conversely, equicontinuity without pointwise relative compactness permits the values of the functions to escape every compact subset of the codomain.
These conditions identify the two independent forms of control required by the theorem. Pointwise relative compactness restricts the possible function values, while equicontinuity prevents those values from changing at increasingly fine spatial scales.