Existing in Mathlib · Analysis and topology

Arzelà–Ascoli Theorem

On a compact domain, a pointwise equicontinuous family of bounded continuous functions whose values remain in one compact set has compact closure for uniform convergence.

compact domain
bounded continuous functions
equicontinuity
compact range
uniform convergence

Exact theorem

Exact Mathlib statement

theorem BoundedContinuousFunction.arzela_ascoli {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [T2Space β] (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β)) (in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) : IsCompact (closure A)

The theorem at a glance

Arzelà–Ascoli Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory analysis poster shows a tightly controlled family of continuous curves on a compact domain, finite sampling grids, compact value cells, and a uniformly convergent subsequence entering the compact closure. — Equicontinuity and a common compact range make the closure of the function family compact in the uniform topology.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A compact source strip supports a family of uniformly controlled curves whose finite sample beads fit inside a compact target mosaic and gather into a compact closure. — Common local oscillation control and compact range discretize the whole function family.

Why it matters

A mathematical landmark

Arzelà–Ascoli turns local control of an entire function family into global compactness. It is one of analysis's fundamental tools for extracting uniformly convergent subsequences and proving existence by compactness.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's bounded-continuous-function Arzelà–Ascoli theorem: on a compact domain, a pointwise equicontinuous family whose values stay in a fixed compact subset of a Hausdorff pseudometric codomain has compact closure in the uniform topology. It does not assert that the original family is closed or compact, omit the common compact-range condition, or state a pointwise-topology result.

The selected conclusion is compactness of closure A, not compactness of A without a closedness hypothesis.
Every function value must remain in one recorded compact subset s of the codomain.
The topology is the bounded-continuous-function uniform topology, not merely pointwise convergence.

Source and local evidence

Where the theorem comes from

Existing declaration: BoundedContinuousFunction.arzela_ascoli in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.arzela-ascoli-theorem.v001
Source: Open the pinned upstream reference