Theorems · Theorem · functional analysis
IsSelfAdjoint.isSymmetric
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
[inst_3 : CompleteSpace E] {A : E →L[𝕜] E}, IsSelfAdjoint A → (↑A).IsSymmetricEvery self-adjoint operator on an inner product space is symmetric.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 194 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.coeproof · cited by 62,936
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- ContinuousLinearMapstatement and proof · cited by 5,352
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- CompleteSpacestatement and proof · cited by 2,532
- Inner.innerproof · cited by 1,089
- IsSelfAdjointstatement and proof · cited by 545
- ContinuousLinearMap.toLinearMapstatement · cited by 528
- LinearMap.IsSymmetricstatement · cited by 121
- ContinuousLinearMap.adjoint_inner_rightproof · cited by 8
Cited by4
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.isSelfAdjoint_iff_isSymmetricproof · cited by 3
- ContinuousLinearMap.isStarNormal_iff_norm_eq_adjointproof · cited by 2
- IsSelfAdjoint.eq_smul_self_of_isLocalExtrOnproof · cited by 1
- IsSelfAdjoint.linearly_dependent_of_isLocalExtrOnproof · cited by 1