Theorems · Theorem · functional analysis
inner_self_eq_zero
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{x : E}, inner 𝕜 x x = 0 ↔ x = 0- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 18 results in Mathlib
- Foundations
- Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Inner.innerstatement · cited by 1,089
- inner_self_eq_norm_sq_to_Kproof · cited by 72
- norm_eq_zeroproof · cited by 43
- sq_eq_zero_iffproof · cited by 11
- RCLike.ofReal_eq_zeroproof · cited by 6
Cited by18
Results whose statement or proof uses this declaration.
- ext_inner_rightproof · cited by 13
- ext_inner_leftproof · cited by 11
- inner_self_ne_zeroproof · cited by 7
- Submodule.inf_orthogonal_eq_botproof · cited by 6
- Submodule.eq_starProjection_of_mem_of_inner_eq_zeroproof · cited by 6
- Submodule.top_orthogonal_eq_botproof · cited by 6
- ContinuousLinearMap.ker_adjoint_comp_selfproof · cited by 4
- LinearMap.IsSymmetric.inner_map_self_eq_zeroproof · cited by 2
- inner_map_self_eq_zeroproof · cited by 1
- Matrix.posDef_gram_of_linearIndependentproof · cited by 1
- LinearMap.ker_le_ker_of_rangeproof · cited by 1
- EuclideanGeometry.Sphere.secondInter_eq_self_iffproof · cited by 1