Theorems · Theorem · functional analysis
WeakDual.isBounded_toStrongDual_preimage_iff_isBounded
∀ {𝕜 : Type u_1} {E : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : SeminormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {s : Set (StrongDual 𝕜 E)},
Bornology.IsBounded (⇑WeakDual.toStrongDual ⁻¹' s) ↔ Bornology.IsBounded sA set in WeakDual 𝕜 E is bounded iff its image in StrongDual 𝕜 E is bounded.
- Defined in
- Mathlib.Analysis.Normed.Module.WeakDual
- 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.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement and proof · cited by 53,352
- RingHom.idstatement · cited by 18,349
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Set.preimagestatement · cited by 4,946
- LinearEquivstatement · cited by 3,317
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- StrongDualstatement and proof · cited by 459
- Bornology.IsBoundedstatement · cited by 293
- WeakDualstatement · cited by 103
- WeakDual.toStrongDualstatement · cited by 20
Cited by2
Results whose statement or proof uses this declaration.
- WeakDual.isBounded_closedBallproof · cited by 3
- WeakDual.isBounded_polarproof · cited by 3