Theorems · Theorem · convex and discrete geometry
IsCompact.extremePoints_nonempty
∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [T2Space E]
[IsTopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E},
IsCompact s → s.Nonempty → (Set.extremePoints ℝ s).NonemptyKrein-Milman lemma: In an LCTVS, any nonempty compact set has an extreme point.
- Defined in
- Mathlib.Analysis.Convex.KreinMilman
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites51
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Set.Elemproof · cited by 7,166
- Set.ofPredproof · cited by 6,101
- Set.Nonemptystatement and proof · cited by 2,627
- IsClosedproof · cited by 1,639
- IsTopologicalAddGroupstatement and proof · cited by 1,394
- T2Spacestatement and proof · cited by 1,351
Cited by2
Results whose statement or proof uses this declaration.
- surjOn_extremePoints_imageproof · cited by 0
- closure_convexHull_extremePointsproof · cited by 0