Mathlib theorem · Existing formal mathematics
ProofAtlas

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.

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.

Equicontinuity and a common compact range make the closure of the function family compact in the uniform topology. Explanatory diagram.
Detailed visual description

The poster separates the compact-domain and compact-range hypotheses, the shared local oscillation control, finite discretization of domain and range, total boundedness, closure, and compactness. A selected curve sequence converges uniformly rather than only pointwise.

Statement structure

Statement and scope

Statement map for Arzelà–Ascoli TheoremA pointwise equicontinuous bounded-continuous-function family on a compact domain, with values in a common compact set, has compact closure in the uniform topology. Claim boundary: 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 pinned upstream declaration is BoundedContinuousFunction.arzela_ascoli. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingA pointwiseequicontinuousbounded-continuous-functionfamily on a compactdomain, with values in acommon compact set, hascompact closure in theuniform topology.Claim boundaryThis page indexesMathlib'sbounded-continuous-functionArzelà–Ascoli theorem:on a compact domain, apointwise equicontinuousfamily whose values stayin a fixed compactsubset of a Hausdorffpseudometric codomainhas compact closure inthe uniform topology. Itdoes not assert that theoriginal family isclosed or compact, omitthe common compact-rangecondition, or state apointwise-topologyresult.Pinned declarationmathlib ·BoundedContinuousFunction.arzela_ascoliStatement map for Arzelà–Ascoli TheoremA pointwise equicontinuous bounded-continuous-function family on a compact domain, with values in a common compact set, has compact closure in the uniform topology. Claim boundary: 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 pinned upstream declaration is BoundedContinuousFunction.arzela_ascoli. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingA pointwiseequicontinuousbounded-continuous-functionfamily on a compactdomain, with values in acommon compact set, hascompact closure in theuniform topology.Claim boundaryThis page indexesMathlib'sbounded-continuous-functionArzelà–Ascoli theorem:on a compact domain, apointwise equicontinuousfamily whose values stayin a fixed compactsubset of a Hausdorffpseudometric codomainhas compact closure inthe uniform topology. Itdoes not assert that theoriginal family isclosed or compact, omitthe common compact-rangecondition, or state apointwise-topologyresult.Pinned declarationmathlib ·BoundedContinuousFunction.arzela_ascoli

Read the exact Mathlib declaration

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

Common local oscillation control and compact range discretize the whole function family. Explanatory diagram.
Detailed visual description

A wide mathematical field shows many distinct continuous curves sharing one modulus-like local control across a compact domain. Finite sampling windows and a finite compact-range net compress the family into bounded data, leading to one enclosed closure region.

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.

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