Theorems · Theorem · operator theory
LinearMap.IsIdempotentElem.isSymmetric_iff_orthogonal_range
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{T : E →ₗ[𝕜] E}, IsIdempotentElem T → (T.IsSymmetric ↔ T.rangeᗮ = T.ker)- Cited by
- 4 results in Mathlib
- Foundations
- Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- LinearMapstatement and proof · cited by 10,215
- Submodulestatement · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- LinearMap.rangestatement and proof · cited by 893
- LinearMap.kerstatement and proof · cited by 848
- Submodule.orthogonalstatement and proof · cited by 257
- IsIdempotentElemstatement and proof · cited by 217
- LinearMap.IsSymmetricstatement and proof · cited by 121
- Iff.eqproof · cited by 14
Cited by4
Results whose statement or proof uses this declaration.
- LinearMap.IsSymmetricProjection.ext_iffproof · cited by 2
- LinearMap.IsSymmetricProjection.hasOrthogonalProjection_rangeproof · cited by 1
- LinearMap.isSymmetricProjection_iff_eq_coe_starProjection_rangeproof · cited by 1
- ContinuousLinearMap.IsIdempotentElem.TFAEproof · cited by 0