Theorems · Theorem · functional analysis
ConvexOn.univ_sSup_of_countable_affine_eq
∀ {𝕜 : Type u_1} {E : Type u_2} {φ : E → ℝ} [inst : RCLike 𝕜] [inst_1 : TopologicalSpace E] [inst_2 : AddCommGroup E]
[inst_3 : Module ℝ E] [inst_4 : Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]
[LocallyConvexSpace ℝ E] [HereditarilyLindelofSpace E],
LowerSemicontinuous φ →
ConvexOn ℝ Set.univ φ →
∃ 𝓕', 𝓕'.Countable ∧ sSup 𝓕' = φ ∧ ∀ f ∈ 𝓕', f ≤ φ ∧ ∃ l c, f = ⇑RCLike.re ∘ ⇑l + Function.const E cThe countable version of univ_sSup_affine_eq.
- Defined in
- Mathlib.Analysis.Convex.Approximation
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites43
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · 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
- RingHom.idstatement and proof · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- Set.Elemproof · cited by 7,166
- Set.ofPredproof · cited by 6,101
- ContinuousLinearMapstatement and proof · cited by 5,352
- Set.univstatement and proof · cited by 3,945
- IsScalarTowerstatement and proof · cited by 3,896
Cited by1
Results whose statement or proof uses this declaration.
- ConvexOn.real_univ_sSup_of_countable_affine_eqproof · cited by 0