Theorems · Theorem · functional analysis
Submodule.sub_starProjection_mem_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} [inst_3 : K.HasOrthogonalProjection] (v : E), v - K.starProjection v ∈ KᗮThe difference of v from its orthogonal projection onto K is in Kᗮ.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 178 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.
- 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
- Submodule.orthogonalstatement · cited by 257
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
- Submodule.starProjectionstatement · cited by 92
- inner_eq_zero_symmproof · cited by 12
- Submodule.starProjection_inner_eq_zeroproof · cited by 8
Cited by8
Results whose statement or proof uses this declaration.
- Submodule.starProjection_orthogonal_valproof · cited by 7
- Submodule.orthogonalProjectionOnto_eq_zero_iffproof · cited by 3
- EuclideanGeometry.orthogonalProjection_mem_orthogonalproof · cited by 3
- Submodule.orthogonalProjectionOnto_starProjection_of_leproof · cited by 3
- Submodule.sup_orthogonal_inf_of_hasOrthogonalProjectionproof · cited by 2
- Submodule.orthogonalProjectionOnto_orthogonalstatement · cited by 2
- Submodule.exists_add_mem_mem_orthogonalproof · cited by 1
- Submodule.orthogonalProjection_orthogonalstatement · cited by 0