Theorems · Theorem · operator theory
LinearMap.IsIdempotentElem.isPositive_iff_isSymmetric
∀ {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.IsPositive ↔ T.IsSymmetric)An idempotent linear map is positive iff it is symmetric.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 109 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.
- DFunLike.coeproof · cited by 62,936
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- LinearMapstatement and proof · cited by 10,215
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Inner.innerproof · cited by 1,089
- RCLike.reproof · cited by 319
- IsIdempotentElemstatement and proof · cited by 217
- LinearMap.IsSymmetricstatement and proof · cited by 121
- IsIdempotentElem.eqproof · cited by 42
- LinearMap.IsPositivestatement and proof · cited by 39
Cited by2
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.IsIdempotentElem.isPositive_iff_isSelfAdjointproof · cited by 2
- LinearMap.IsSymmetricProjection.isPositiveproof · cited by 1