Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Analysis

Krein–Milman Theorem

In a real Hausdorff locally convex topological vector space, a compact convex set is completely recovered by taking all convex combinations of its extreme points and then taking the closure. The closure matters: this statement does not reduce the set to a finite polytope or promise a finite representation of each point.

Exact theorem

Exact Mathlib statement

theorem closure_convexHull_extremePoints {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E] [IsTopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E} (hscomp : IsCompact s) (hAconv : Convex ℝ s) : closure (convexHull ℝ (s.extremePoints ℝ)) = s

The theorem at a glance

Krein–Milman Theorem at a glance

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

Compact convex sets in real Hausdorff locally convex spaces are the closed convex hulls of their extreme points. Explanatory diagram.
Detailed visual description

The central set is a smooth convex oval: all displayed chords remain inside it, a continuous boundary glow avoids a finite-generator implication, and a thin halo records closure. Four panels use convex bodies throughout and follow the checked route: the easy inclusion, a hypothetical point outside the closed hull, continuous-linear-functional separation, and a supporting exposed face whose extreme point already belongs to the closed hull.

Statement structure

Statement and scope

Statement map for Krein–Milman TheoremEvery compact convex set in a real Hausdorff locally convex topological vector space is the closure of the convex hull of its extreme points. Claim boundary: This page indexes Mathlib's Krein–Milman theorem in a real Hausdorff locally convex topological vector space: every compact convex set s equals closure (convexHull ℝ (s.extremePoints ℝ)). No finite-dimensional hypothesis is present, and the closure is part of the formal conclusion. The set may be empty. The selected declaration does not supply a finite convex representation, a bound on the number of extreme points, or the finite-dimensional strengthening in which closure can sometimes be removed. The pinned upstream declaration is closure_convexHull_extremePoints. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingEvery compact convex setin a real Hausdorfflocally convextopological vector spaceis the closure of theconvex hull of itsextreme points.Claim boundaryThis page indexesMathlib's Krein–Milmantheorem in a realHausdorff locally convextopological vectorspace: every compactconvex set s equalsclosure (convexHull ℝ(s.extremePoints ℝ)). Nofinite-dimensionalhypothesis is present,and the closure is partof the formalconclusion. The set maybe empty. The selecteddeclaration does notsupply a finite convexrepresentation, a boundon the number of extremepoints, or thefinite-dimensionalstrengthening in whichclosure can sometimes beremoved.Pinned declarationmathlib ·closure_convexHull_extremePointsStatement map for Krein–Milman TheoremEvery compact convex set in a real Hausdorff locally convex topological vector space is the closure of the convex hull of its extreme points. Claim boundary: This page indexes Mathlib's Krein–Milman theorem in a real Hausdorff locally convex topological vector space: every compact convex set s equals closure (convexHull ℝ (s.extremePoints ℝ)). No finite-dimensional hypothesis is present, and the closure is part of the formal conclusion. The set may be empty. The selected declaration does not supply a finite convex representation, a bound on the number of extreme points, or the finite-dimensional strengthening in which closure can sometimes be removed. The pinned upstream declaration is closure_convexHull_extremePoints. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingEvery compact convex setin a real Hausdorfflocally convextopological vector spaceis the closure of theconvex hull of itsextreme points.Claim boundaryThis page indexesMathlib's Krein–Milmantheorem in a realHausdorff locally convextopological vectorspace: every compactconvex set s equalsclosure (convexHull ℝ(s.extremePoints ℝ)). Nofinite-dimensionalhypothesis is present,and the closure is partof the formalconclusion. The set maybe empty. The selecteddeclaration does notsupply a finite convexrepresentation, a boundon the number of extremepoints, or thefinite-dimensionalstrengthening in whichclosure can sometimes beremoved.Pinned declarationmathlib ·closure_convexHull_extremePoints

Read the exact Mathlib declaration

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

The compact convex set K is the closure of the convex hull of ext K; the theorem does not assert that extreme points are dense in the boundary. Explanatory scientific diagram.
Detailed visual description

Representative markers indicate extreme points at selected boundary locations without identifying ext K with the whole boundary. The interior field represents convex combinations from the full extreme-point set, and the separate halo represents closure. The markers are schematic, not a finite enumeration.

Why it matters

A mathematical landmark

Krein–Milman is a cornerstone of infinite-dimensional convexity: it says that compact convex structure is controlled by extreme points, while the necessary closure records the genuinely topological content. Mathlib's proof exposes the central Hahn–Banach separation and extreme-face architecture.

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 Krein–Milman theorem in a real Hausdorff locally convex topological vector space: every compact convex set s equals closure (convexHull ℝ (s.extremePoints ℝ)). No finite-dimensional hypothesis is present, and the closure is part of the formal conclusion. The set may be empty. The selected declaration does not supply a finite convex representation, a bound on the number of extreme points, or the finite-dimensional strengthening in which closure can sometimes be removed.

Source and local evidence

Where the theorem comes from

Existing declaration
closure_convexHull_extremePoints in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.krein-milman-theorem.v001
Source
Open the pinned upstream reference