Theorems · Theorem · functional analysis
Submodule.isClosed_orthogonal
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E), IsClosed ↑KᗮThe orthogonal complement of any submodule K is closed.
- Cited by
- 4 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.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setproof · cited by 53,352
- TopologicalSpaceproof · cited by 24,529
- NormedAddCommGroupstatement and proof · cited by 15,752
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- iInfproof · cited by 1,690
- IsClosedstatement and proof · cited by 1,639
- Set.iInterproof · cited by 1,084
- LinearMap.kerproof · cited by 848
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.orthogonal_orthogonal_eq_closureproof · cited by 3
- ClosedSubmodule.orthogonal_closure'statement · cited by 1
- LinearPMap.adjoint_isClosedproof · cited by 1
- ClosedSubmodule.orthogonal_closure''proof · cited by 0