Theorems · Theorem · functional analysis
Submodule.orthogonalProjectionOnto_mem_subspace_eq_self
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{K : Submodule 𝕜 E} [inst_3 : K.HasOrthogonalProjection] (v : ↥K), K.orthogonalProjectionOnto ↑v = vThe orthogonal projection sends elements of K to themselves.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · 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_selfproof · cited by 996
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
- Submodule.orthogonalProjectionOntostatement · cited by 103
- inner_zero_leftproof · cited by 59
Cited by8
Results whose statement or proof uses this declaration.
- MeasureTheory.condExpL2_indicator_of_measurableproof · cited by 3
- stereo_right_invproof · cited by 1
- ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjectionproof · cited by 0
- Submodule.orthogonalProjection_mem_subspace_eq_selfproof · cited by 0
- Submodule.starProjection_comp_starProjection_eq_zero_iffproof · cited by 0
- LinearIsometry.extend_applyproof · cited by 0
- Submodule.starProjection_mem_subspace_eq_selfproof · cited by 0
- OrthogonalFamily.projection_directSum_coeAddHomproof · cited by 0