Existing in Mathlib · Functional analysis

Banach–Alaoglu Theorem

Over a proper nontrivially normed field, every operator-norm closed ball of continuous linear functionals becomes compact when the dual is viewed with the weaker weak-* topology.

continuous linear functionals
weak-* topology
operator-norm closed balls
compactness
proper scalar fields
functional analysis

Exact theorem

Exact Mathlib statement

theorem WeakDual.isCompact_closedBall {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] (x' : StrongDual 𝕜 E) (r : ℝ) : IsCompact (WeakDual.toStrongDual ⁻¹' Metric.closedBall x' r)

The theorem at a glance

Banach–Alaoglu theorem at a glance

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

An ivory Banach–Alaoglu poster places the proper-scalar-field hypothesis and a seminormed additive commutative group E carrying a NormedSpace 𝕜 E structure above a large closed dual ball whose functional traces pass into a receding weak-star coordinate lattice, followed by the bounded-and-closed compactness route. — An operator-norm closed ball in the continuous dual is compact for the weak-* topology when the scalar field is proper.

Statement structure

Statement and scope

Read the exact Mathlib declaration

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

Scientific figure illustrating: Probe points feed a gold-bounded family of functional strands into an indefinitely receding weak-star coordinate field. — Point evaluations organize one operator-norm bounded family inside a receding weak-star coordinate field.

Why it matters

A mathematical landmark

Banach–Alaoglu is a cornerstone of functional analysis: it converts an operator-norm bound into compactness after passing to the weak-* topology. Mathlib's selected endpoint makes the topology change, the proper scalar-field assumption, and the absence of completeness or separability assumptions on E explicit.

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 closed-ball form of the Banach–Alaoglu theorem. For a proper nontrivially normed field 𝕜 and a seminormed additive commutative group E carrying a NormedSpace 𝕜 E structure, the inverse image in WeakDual 𝕜 E of an operator-norm closed ball in StrongDual 𝕜 E is compact in the weak-* topology. The selected declaration does not claim operator-norm compactness or sequential compactness, does not cover arbitrary nonproper scalar fields, and requires neither completeness nor separability of E.

The compactness conclusion is for the weak-* topology on WeakDual 𝕜 E, not the operator-norm topology on StrongDual 𝕜 E.
The selected declaration assumes that the nontrivially normed scalar field 𝕜 is a ProperSpace; it does not state the result for an arbitrary nonproper scalar field.
The declaration does not assert sequential compactness. No separability or completeness assumption on E appears in this endpoint.
ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: WeakDual.isCompact_closedBall in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.banach-alaoglu-theorem.v001
Source: Open the pinned upstream reference