Theorems · Theorem · functional analysis
SeparatingDual.completeSpace_continuousMultilinearMap_iff
∀ (𝕜 : Type u_1) (F : Type u_3) [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {ι : Type u_4} [Finite ι] {M : ι → Type u_5} [inst_4 : (i : ι) → NormedAddCommGroup (M i)]
[inst_5 : (i : ι) → NormedSpace 𝕜 (M i)] [∀ (i : ι), SeparatingDual 𝕜 (M i)] {m : (i : ι) → M i},
(∀ (i : ι), m i ≠ 0) → (CompleteSpace (ContinuousMultilinearMap 𝕜 M F) ↔ CompleteSpace F)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Finitestatement and proof · cited by 3,029
- CompleteSpacestatement and proof · cited by 2,532
- ContinuousMultilinearMapstatement and proof · cited by 1,016
- SeparatingDualstatement and proof · cited by 24
- SeparatingDual.completeSpace_of_completeSpace_continuousMultilinearMapproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- ContinuousMultilinearMap.integral_applyproof · cited by 1