Theorems · Theorem · functional analysis
Submodule.orthogonal_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) [K.HasOrthogonalProjection], Kᗮᗮ = KIf K admits an orthogonal projection, then the orthogonal complement of its orthogonal
complement is itself.
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realproof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement and proof · cited by 7,192
- Norm.normproof · cited by 5,413
- Algebra.algebraMapproof · cited by 4,706
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- add_zeroproof · cited by 2,707
- zero_addproof · cited by 2,366
- Inner.innerproof · cited by 1,089
- RCLike.ofRealproof · cited by 350
- Submodule.orthogonalstatement and proof · cited by 257
Cited by14
Results whose statement or proof uses this declaration.
- Submodule.orthogonalDecomposition_applyproof · cited by 6
- ContinuousLinearMap.orthogonal_rangeproof · cited by 3
- ClosedSubmodule.orthogonal_orthogonal_eqproof · cited by 3
- Submodule.orthogonal_orthogonal_eq_closureproof · cited by 3
- EuclideanGeometry.Sphere.orthRadius_le_orthRadius_iffproof · cited by 2
- Submodule.orthogonal_le_orthogonal_iffproof · cited by 2
- ContinuousLinearMap.mem_invtSubmodule_adjoint_iffproof · cited by 1
- stereographic_apply_negproof · cited by 1
- Submodule.orthogonalComplement_eq_orthogonalComplementproof · cited by 0
- Submodule.le_orthogonal_iff_le_orthogonalproof · cited by 0
- EuclideanGeometry.Sphere.orthogonalProjection_orthRadius_centerproof · cited by 0
- LinearIsometry.extend_applyproof · cited by 0