Theorems · Theorem · functional analysis
ClosedSubmodule.sup_orthogonal
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
[CompleteSpace E] (K₁ K₂ : ClosedSubmodule 𝕜 E), K₁ᗮ ⊔ K₂ᗮ = (K₁ ⊓ K₂)ᗮThe sup of two orthogonal subspaces equals the subspace orthogonal to the inf.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 183 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
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- CompleteSpacestatement and proof · cited by 2,532
- ClosedSubmodulestatement and proof · cited by 123
- ClosedSubmodule.orthogonalstatement and proof · cited by 32
- ClosedSubmodule.orthogonal_orthogonal_eqproof · cited by 3
- ClosedSubmodule.inf_orthogonalproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- ClosedSubmodule.symplComp_infproof · cited by 0