Theorems · Theorem · functional analysis
Submodule.sup_orthogonal_of_hasOrthogonalProjection
∀ {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], K ⊔ Kᗮ = ⊤If K admits an orthogonal projection, then K and Kᗮ span the whole space.
- Cited by
- 2 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.
Cites10
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
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- le_topproof · cited by 411
- Submodule.orthogonalstatement and proof · cited by 257
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
- inf_of_le_leftproof · cited by 186
- Submodule.sup_orthogonal_inf_of_hasOrthogonalProjectionproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Submodule.orthogonal_eq_bot_iffproof · cited by 7
- LinearMap.IsSymmetric.isFinitelySemisimpleproof · cited by 1