Theorems · Theorem · functional analysis
LinearMap.polar_eq_biInter_preimage
∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NormedCommRing 𝕜] [inst_1 : AddCommMonoid E]
[inst_2 : AddCommMonoid F] [inst_3 : Module 𝕜 E] [inst_4 : Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E),
B.polar s = ⋂ x ∈ s, ⇑(B x) ⁻¹' Metric.closedBall 0 1- Defined in
- Mathlib.Analysis.LocallyConvex.Polar
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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 · cited by 25,697
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Set.preimagestatement · cited by 4,946
- Set.extproof · cited by 2,266
- Set.iInterstatement · cited by 1,084
- Metric.closedBallstatement · cited by 704
- NormedCommRingstatement and proof · cited by 218
Cited by2
Results whose statement or proof uses this declaration.
- LinearMap.polar_isClosedproof · cited by 0
- LinearMap.polar_AbsConvexproof · cited by 0