Theorems · Theorem · functional analysis
ClosedSubmodule.orthogonal_closure
∀ {𝕜 : Type u_4} {E : Type u_5} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E), (↑K.closure)ᗮ = KᗮThe orthogonal complement of the closure of a submodule (as a Submodule) is equal to
the orthogonal complement.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Submodulestatement and proof · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Submodule.orthogonalstatement and proof · cited by 257
- ClosedSubmodule.toSubmodulestatement and proof · cited by 51
- Submodule.topologicalClosureproof · cited by 50
- Submodule.closurestatement and proof · cited by 16
- Submodule.orthogonal_closureproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- ClosedSubmodule.orthogonal_closure'proof · cited by 1