Theorems · Theorem · functional analysis
Submodule.orthogonal_closure
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E), K.topologicalClosureᗮ = KᗮThe closure of a submodule has the same orthogonal complement and the submodule itself.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- le_antisymmproof · cited by 2,068
- continuous_id'proof · cited by 295
- continuous_constproof · cited by 278
- Submodule.orthogonalstatement and proof · cited by 257
- closure_minimalproof · cited by 94
- isClosed_eqproof · cited by 71
- Submodule.topologicalClosurestatement and proof · cited by 50
- Continuous.innerproof · cited by 19
Cited by3
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.orthogonal_rangeproof · cited by 3
- ClosedSubmodule.orthogonal_closureproof · cited by 1
- Submodule.orthogonal_closure'proof · cited by 0