General topology Uniform and proximity structures Theorem

Arzelà–Ascoli theorem

Builds on Equicontinuity Uniform metric on bounded functions Compact space Compact iff complete and totally bounded

Statement

Let $X$ be a compact metric space and let $C(X)$ be the set of continuous functions $X \to \mathbb{R}$ , with the uniform metric $d_\infty(f, g) = \sup_{x \in X} |f(x) - g(x)|$ . Let $\mathcal{F} \subseteq C(X)$ . Then the closure of $\mathcal{F}$ in $C(X)$ is compact if and only if $\mathcal{F}$ is equicontinuous and uniformly bounded. Here uniformly bounded means that there is a real $M$ with $|f(x)| \le M$ for all $f \in \mathcal{F}$ and $x \in X$ .

Remarks

A sequential form: each uniformly bounded and equicontinuous sequence of functions $[a, b] \to \mathbb{R}$ has a subsequence that converges uniformly on $[a, b]$ . The proof shows that $\mathcal{F}$ is totally bounded in the complete space $C(X)$ , and then uses compact iff complete and totally bounded. For example, the functions $f \colon [0, 1] \to [0, 1]$ that are Lipschitz with constant $1$ form a compact set in $C([0, 1])$ . The theorem is a main tool for existence proofs, for example in the Peano existence theorem for ordinary differential equations.

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

Element definition
Extended real number line object
Proposition concept
Real numbers object
Set bounded above definition
Uniform metric on bounded functions example
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Supremum definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Topological space definition
Sequential compactness definition
Function definition
Subspace topology definition
Preimage of a set definition
Restriction of a function definition
Metric space definition
Subspace metric definition
Convergence of a sequence in a metric space definition
Open ball definition
Diameter of a set definition
Metric topology definition
Epsilon characterisation of Cauchy sequences statement
Totally bounded set definition
Open set definition
Complete metric space definition
Open cover definition
Open sets characterised by neighbourhoods statement
Open sets of a metric space statement
Compact space definition
Continuity at a point of a topological space definition
Compact set definition
Lebesgue number lemma statement
Epsilon-delta continuity in metric spaces statement
Compactness and sequential compactness agree in metric spaces statement
Equicontinuity definition
Compact iff complete and totally bounded statement
Arzelà–Ascoli theorem statement

Where this leads

14 pages build on this one.

Tag	`0381S8Z5P0`
URI	`m2://0381S8Z5P0`
Aliases	`ascoli-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	54C35
Status	This copy of the site has no build data. Run `m2 status 0381S8Z5P0`.
Uses	Equicontinuity, Uniform metric on bounded functions, Compact space, Compact iff complete and totally bounded
Same as	Wikidata Q1477053 Wikipedia Arzelà–Ascoli theorem

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