Arzelà-Ascoli Theorem

The Arzelà-Ascoli Theorem characterizes compact sets of continuous functions. A set of functions is compact (has convergent subsequences) if and only if it is closed, pointwise bounded, and equicontinuous. This is the function-space analogue of the Heine-Borel theorem and is essential in PDE theory and functional analysis.

Statement

Theorem7.1Arzelà-Ascoli Theorem

Let $(f_n)$ be a sequence of functions on $[a, b]$ . Then $(f_n)$ has a uniformly convergent subsequence if and only if:

$(f_n)$ is uniformly bounded: there exists $M$ such that $|f_n(x)| \leq M$ for all $n$ and $x \in [a, b]$ .
$(f_n)$ is equicontinuous: for every $\epsilon > 0$ , there exists $\delta > 0$ such that $|f_n(x) - f_n(y)| < \epsilon$ whenever $|x - y| < \delta$ , for all $n$ .

RemarkEquicontinuity

Equicontinuity means all $f_n$ have the same "modulus of continuity" — the same $\delta$ works for all $n$ simultaneously. This prevents the functions from oscillating wildly.

Applications

ExampleLipschitz functions

The set of all Lipschitz functions on $[0, 1]$ with Lipschitz constant $\leq 1$ and $|f(x)| \leq 1$ is compact by Arzelà-Ascoli. Every sequence has a uniformly convergent subsequence.

ExampleSolutions of ODEs

Arzelà-Ascoli is used to prove existence of solutions to ODEs via Picard iteration: the sequence of approximate solutions is equicontinuous (by the Lipschitz condition on the ODE), hence has a convergent subsequence.

Summary

Arzelà-Ascoli characterizes compactness in function spaces:

Compactness $\Leftrightarrow$ uniform boundedness + equicontinuity.
Generalization of Bolzano-Weierstrass to functions.
Applications: PDEs, ODEs, calculus of variations.

See Uniform Convergence and Metric Spaces.