Theorems · Theorem · functional analysis
Submodule.orthogonalProjectionOnto_comp_subtypeL_eq_zero_iff
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{U V : Submodule 𝕜 E} [inst_3 : U.HasOrthogonalProjection], U.orthogonalProjectionOnto ∘SL V.subtypeL = 0 ↔ U ⟂ VThe projection into U from V is the zero map if and only if U and V are orthogonal.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement and proof · cited by 7,192
- ContinuousLinearMapstatement · cited by 5,352
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Inner.innerproof · cited by 1,089
- sub_zeroproof · cited by 938
- ContinuousLinearMap.compstatement and proof · cited by 709
- DFunLike.congr_funproof · cited by 288
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
Cited by2
Results whose statement or proof uses this declaration.
- Submodule.orthogonalProjection_comp_subtypeL_eq_zero_iffproof · cited by 0
- Submodule.starProjection_comp_starProjection_eq_zero_iffproof · cited by 0