ArzelàAscolitheorema

The Arzelà–Ascoli theorem is a compactness criterion for families of continuous functions. Let X be a compact metric space and F a family of real-valued continuous functions on X.

Suppose F is uniformly bounded: there exists M such that |f(x)| ≤ M for all f ∈ F and x ∈ X, and equicontinuous: for every ε > 0 there exists δ > 0 such that for all f ∈ F and all x,y ∈ X with d(x,y) < δ, we have |f(x) - f(y)| < ε. Then F is relatively compact in (C(X,R), ||·||∞). Equivalently, every sequence in F has a subsequence converging uniformly to a continuous function.

More generally, if F ⊂ C(X,Y) where Y is a complete metric space, and if F is equicontinuous and for each x ∈ X the set {f(x): f ∈ F} has compact closure in Y, then F is relatively compact in C(X,Y) with the sup norm.

Remarks: The theorem is a fundamental tool in analysis and partial differential equations, used to extract convergent subsequences from families of functions and to prove existence results for solutions of integral and differential equations.

History: the theorem is named after Cesàro, Arzelà, and Ascoli, who contributed to its development in the late 19th century.