Theorems · Theorem · operator theory
LinearMap.IsSymmetric.inner_map_self_eq_zero
∀ {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}, T.IsSymmetric → ((∀ (x : E), inner 𝕜 (T x) x = 0) ↔ T = 0)A symmetric linear map T is zero if and only if ⟪T x, x⟫_ℝ = 0 for all x.
See inner_map_self_eq_zero for the complex version without the symmetric assumption.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 170 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 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
- Nat.cast_zeroproof · cited by 1,870
- Inner.innerstatement and proof · cited by 1,089
- LinearMap.IsSymmetricstatement and proof · cited by 121
- RCLike.Iproof · cited by 100
- inner_zero_leftproof · cited by 59
- inner_self_eq_zeroproof · cited by 18
Cited by2
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.isStarNormal_iff_norm_eq_adjointproof · cited by 2
- LinearMap.IsPositive.ne_zero_iffproof · cited by 1