Theorems · Theorem · functional analysis
Submodule.orthogonal_eq_bot_iff
∀ {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.HasOrthogonalProjection], Kᗮ = ⊥ ↔ K = ⊤- Cited by
- 7 results in Mathlib
- Foundations
- Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Top.topstatement and proof · cited by 9,680
- Submodulestatement and proof · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Submodule.orthogonalstatement and proof · cited by 257
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
- sup_commproof · cited by 165
- bot_sup_eqproof · cited by 32
- Submodule.top_orthogonal_eq_botproof · cited by 6
- Submodule.sup_orthogonal_of_hasOrthogonalProjectionproof · cited by 2
Cited by7
Results whose statement or proof uses this declaration.
- LinearMap.IsSymmetric.iSup_iInf_eq_top_of_commuteproof · cited by 1
- LinearIsometryEquiv.reflections_generate_dim_auxproof · cited by 1
- maximal_orthonormal_iff_basis_of_finiteDimensionalproof · cited by 0
- LinearMap.IsSymmetric.directSum_isInternal_of_commuteproof · cited by 0
- RKHS.kerFun_denseproof · cited by 0
- EuclideanGeometry.Sphere.inter_orthRadius_eq_empty_iffproof · cited by 0