Theorems · Theorem · functional analysis
Submodule.starProjection_eq_self_iff
∀ {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}, K.starProjection v = v ↔ v ∈ KA point equals its orthogonal projection if and only if it lies in the subspace.
- Cited by
- 9 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 and proof · 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.starProjectionstatement and proof · cited by 92
- inner_zero_leftproof · cited by 59
Cited by9
Results whose statement or proof uses this declaration.
- Submodule.reflection_eq_self_iffproof · cited by 3
- Submodule.isIdempotentElem_starProjectionproof · cited by 3
- Submodule.norm_orthogonalProjectionOnto_applyproof · cited by 3
- Submodule.range_starProjectionproof · cited by 1
- Submodule.starProjection_topproof · cited by 1
- EuclideanGeometry.orthogonalProjection_apply_memproof · cited by 0
- Submodule.starProjection_apply_eq_zero_iffproof · cited by 0
- OrthogonalFamily.sum_projection_of_mem_iSupproof · cited by 0
- Submodule.starProjection_tendsto_selfproof · cited by 0